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",[583,1129,1131,1149],{"className":1130},[586],[583,1132,1134],{"className":1133},[590],[592,1135,1136],{"xmlns":594},[596,1137,1138,1146],{},[599,1139,1140],{},[1141,1142,1143],"msqrt",{},[618,1144,1145],{},"2",[631,1147,1148],{"encoding":633},"\\sqrt{2}",[583,1150,1152],{"className":1151,"ariaHidden":639},[638],[583,1153,1155,1159],{"className":1154},[643],[583,1156],{"className":1157,"style":1158},[647],"height:1.04em;vertical-align:-0.1328em;",[583,1160,1163],{"className":1161},[652,1162],"sqrt",[583,1164,1166,1215],{"className":1165},[664,665],[583,1167,1169,1212],{"className":1168},[669],[583,1170,1173,1189],{"className":1171,"style":1172},[673],"height:0.9072em;",[583,1174,1178,1182],{"className":1175,"style":1177},[1176],"svg-align","top:-3em;",[583,1179],{"className":1180,"style":1181},[681],"height:3em;",[583,1183,1186],{"className":1184,"style":1185},[652],"padding-left:0.833em;",[583,1187,1145],{"className":1188},[652],[583,1190,1192,1195],{"style":1191},"top:-2.8672em;",[583,1193],{"className":1194,"style":1181},[681],[583,1196,1200],{"className":1197,"style":1199},[1198],"hide-tail","min-width:0.853em;height:1.08em;",[1201,1202,1208],"svg",{"xmlns":1203,"width":1204,"height":1205,"viewBox":1206,"preserveAspectRatio":1207},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","400em","1.08em","0 0 400000 1080","xMinYMin slice",[1209,1210],"path",{"d":1211},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[583,1213,707],{"className":1214},[706],[583,1216,1218],{"className":1217},[669],[583,1219,1222],{"className":1220,"style":1221},[673],"height:0.1328em;",[583,1223],{}," wachsen und sind beschränkt, haben aber keinen rationalen Grenzwert.",[564,1226,1227,1228,1231,1232,1405,1406,1434,1435,1501],{},"Typischer Einsatz sind ",[568,1229,1230],{},"rekursive Folgen"," ",[583,1233,1235,1278],{"className":1234},[586],[583,1236,1238],{"className":1237},[590],[592,1239,1240],{"xmlns":594},[596,1241,1242,1275],{},[599,1243,1244,1256,1259,1262,1266,1272],{},[602,1245,1246,1248],{},[605,1247,607],{},[599,1249,1250,1252,1254],{},[605,1251,612],{},[614,1253,616],{},[618,1255,620],{},[614,1257,1258],{},"=",[605,1260,1261],{},"g",[614,1263,1265],{"stretchy":1264},"false","(",[602,1267,1268,1270],{},[605,1269,607],{},[605,1271,612],{},[614,1273,1274],{"stretchy":1264},")",[631,1276,1277],{"encoding":633},"a_{n+1} = g(a_n)",[583,1279,1281,1346],{"className":1280,"ariaHidden":639},[638],[583,1282,1284,1288,1337,1340,1343],{"className":1283},[643],[583,1285],{"className":1286,"style":1287},[647],"height:0.6389em;vertical-align:-0.2083em;",[583,1289,1291,1294],{"className":1290},[652],[583,1292,607],{"className":1293},[652,656],[583,1295,1297],{"className":1296},[660],[583,1298,1300,1329],{"className":1299},[664,665],[583,1301,1303,1326],{"className":1302},[669],[583,1304,1306],{"className":1305,"style":674},[673],[583,1307,1308,1311],{"style":677},[583,1309],{"className":1310,"style":682},[681],[583,1312,1314],{"className":1313},[686,687,688,689],[583,1315,1317,1320,1323],{"className":1316},[652,689],[583,1318,612],{"className":1319},[652,656,689],[583,1321,616],{"className":1322},[699,689],[583,1324,620],{"className":1325},[652,689],[583,1327,707],{"className":1328},[706],[583,1330,1332],{"className":1331},[669],[583,1333,1335],{"className":1334,"style":714},[673],[583,1336],{},[583,1338],{"className":1339,"style":721},[720],[583,1341,1258],{"className":1342},[725],[583,1344],{"className":1345,"style":721},[720],[583,1347,1349,1353,1357,1361,1401],{"className":1348},[643],[583,1350],{"className":1351,"style":1352},[647],"height:1em;vertical-align:-0.25em;",[583,1354,1261],{"className":1355,"style":1356},[652,656],"margin-right:0.0359em;",[583,1358,1265],{"className":1359},[1360],"mopen",[583,1362,1364,1367],{"className":1363},[652],[583,1365,607],{"className":1366},[652,656],[583,1368,1370],{"className":1369},[660],[583,1371,1373,1393],{"className":1372},[664,665],[583,1374,1376,1390],{"className":1375},[669],[583,1377,1379],{"className":1378,"style":754},[673],[583,1380,1381,1384],{"style":677},[583,1382],{"className":1383,"style":682},[681],[583,1385,1387],{"className":1386},[686,687,688,689],[583,1388,612],{"className":1389},[652,656,689],[583,1391,707],{"className":1392},[706],[583,1394,1396],{"className":1395},[669],[583,1397,1399],{"className":1398,"style":775},[673],[583,1400],{},[583,1402,1274],{"className":1403},[1404],"mclose",": Zuerst Monotonie und Beschränktheit mit vollständiger Induktion zeigen. Erst dann weiß man, dass ein Grenzwert ",[583,1407,1409,1422],{"className":1408},[586],[583,1410,1412],{"className":1411},[590],[592,1413,1414],{"xmlns":594},[596,1415,1416,1420],{},[599,1417,1418],{},[605,1419,607],{},[631,1421,607],{"encoding":633},[583,1423,1425],{"className":1424,"ariaHidden":639},[638],[583,1426,1428,1431],{"className":1427},[643],[583,1429],{"className":1430,"style":804},[647],[583,1432,607],{"className":1433},[652,656]," existiert, und darf ihn aus der Fixpunktgleichung ",[583,1436,1438,1462],{"className":1437},[586],[583,1439,1441],{"className":1440},[590],[592,1442,1443],{"xmlns":594},[596,1444,1445,1459],{},[599,1446,1447,1449,1451,1453,1455,1457],{},[605,1448,607],{},[614,1450,1258],{},[605,1452,1261],{},[614,1454,1265],{"stretchy":1264},[605,1456,607],{},[614,1458,1274],{"stretchy":1264},[631,1460,1461],{"encoding":633},"a = g(a)",[583,1463,1465,1483],{"className":1464,"ariaHidden":639},[638],[583,1466,1468,1471,1474,1477,1480],{"className":1467},[643],[583,1469],{"className":1470,"style":804},[647],[583,1472,607],{"className":1473},[652,656],[583,1475],{"className":1476,"style":721},[720],[583,1478,1258],{"className":1479},[725],[583,1481],{"className":1482,"style":721},[720],[583,1484,1486,1489,1492,1495,1498],{"className":1485},[643],[583,1487],{"className":1488,"style":1352},[647],[583,1490,1261],{"className":1491,"style":1356},[652,656],[583,1493,1265],{"className":1494},[1360],[583,1496,607],{"className":1497},[652,656],[583,1499,1274],{"className":1500},[1404]," bestimmen.",[559,1503,1505],{"id":1504},"beispiele","Beispiele",[564,1507,1508,1601,1602,1819,1820,2121],{},[583,1509,1511,1533],{"className":1510},[586],[583,1512,1514],{"className":1513},[590],[592,1515,1516],{"xmlns":594},[596,1517,1518,1530],{},[599,1519,1520,1526,1528],{},[602,1521,1522,1524],{},[605,1523,607],{},[618,1525,620],{},[614,1527,1258],{},[618,1529,620],{},[631,1531,1532],{"encoding":633},"a_1 = 1",[583,1534,1536,1591],{"className":1535,"ariaHidden":639},[638],[583,1537,1539,1542,1582,1585,1588],{"className":1538},[643],[583,1540],{"className":1541,"style":735},[647],[583,1543,1545,1548],{"className":1544},[652],[583,1546,607],{"className":1547},[652,656],[583,1549,1551],{"className":1550},[660],[583,1552,1554,1574],{"className":1553},[664,665],[583,1555,1557,1571],{"className":1556},[669],[583,1558,1560],{"className":1559,"style":674},[673],[583,1561,1562,1565],{"style":677},[583,1563],{"className":1564,"style":682},[681],[583,1566,1568],{"className":1567},[686,687,688,689],[583,1569,620],{"className":1570},[652,689],[583,1572,707],{"className":1573},[706],[583,1575,1577],{"className":1576},[669],[583,1578,1580],{"className":1579,"style":775},[673],[583,1581],{},[583,1583],{"className":1584,"style":721},[720],[583,1586,1258],{"className":1587},[725],[583,1589],{"className":1590,"style":721},[720],[583,1592,1594,1598],{"className":1593},[643],[583,1595],{"className":1596,"style":1597},[647],"height:0.6444em;",[583,1599,620],{"className":1600},[652],", ",[583,1603,1605,1645],{"className":1604},[586],[583,1606,1608],{"className":1607},[590],[592,1609,1610],{"xmlns":594},[596,1611,1612,1642],{},[599,1613,1614,1626,1628],{},[602,1615,1616,1618],{},[605,1617,607],{},[599,1619,1620,1622,1624],{},[605,1621,612],{},[614,1623,616],{},[618,1625,620],{},[614,1627,1258],{},[1141,1629,1630],{},[599,1631,1632,1634,1636],{},[618,1633,1145],{},[614,1635,616],{},[602,1637,1638,1640],{},[605,1639,607],{},[605,1641,612],{},[631,1643,1644],{"encoding":633},"a_{n+1} = \\sqrt{2 + a_n}",[583,1646,1648,1712],{"className":1647,"ariaHidden":639},[638],[583,1649,1651,1654,1703,1706,1709],{"className":1650},[643],[583,1652],{"className":1653,"style":1287},[647],[583,1655,1657,1660],{"className":1656},[652],[583,1658,607],{"className":1659},[652,656],[583,1661,1663],{"className":1662},[660],[583,1664,1666,1695],{"className":1665},[664,665],[583,1667,1669,1692],{"className":1668},[669],[583,1670,1672],{"className":1671,"style":674},[673],[583,1673,1674,1677],{"style":677},[583,1675],{"className":1676,"style":682},[681],[583,1678,1680],{"className":1679},[686,687,688,689],[583,1681,1683,1686,1689],{"className":1682},[652,689],[583,1684,612],{"className":1685},[652,656,689],[583,1687,616],{"className":1688},[699,689],[583,1690,620],{"className":1691},[652,689],[583,1693,707],{"className":1694},[706],[583,1696,1698],{"className":1697},[669],[583,1699,1701],{"className":1700,"style":714},[673],[583,1702],{},[583,1704],{"className":1705,"style":721},[720],[583,1707,1258],{"className":1708},[725],[583,1710],{"className":1711,"style":721},[720],[583,1713,1715,1719],{"className":1714},[643],[583,1716],{"className":1717,"style":1718},[647],"height:1.04em;vertical-align:-0.2078em;",[583,1720,1722],{"className":1721},[652,1162],[583,1723,1725,1810],{"className":1724},[664,665],[583,1726,1728,1807],{"className":1727},[669],[583,1729,1732,1794],{"className":1730,"style":1731},[673],"height:0.8322em;",[583,1733,1735,1738],{"className":1734,"style":1177},[1176],[583,1736],{"className":1737,"style":1181},[681],[583,1739,1741,1744,1748,1751,1754],{"className":1740,"style":1185},[652],[583,1742,1145],{"className":1743},[652],[583,1745],{"className":1746,"style":1747},[720],"margin-right:0.2222em;",[583,1749,616],{"className":1750},[699],[583,1752],{"className":1753,"style":1747},[720],[583,1755,1757,1760],{"className":1756},[652],[583,1758,607],{"className":1759},[652,656],[583,1761,1763],{"className":1762},[660],[583,1764,1766,1786],{"className":1765},[664,665],[583,1767,1769,1783],{"className":1768},[669],[583,1770,1772],{"className":1771,"style":754},[673],[583,1773,1774,1777],{"style":677},[583,1775],{"className":1776,"style":682},[681],[583,1778,1780],{"className":1779},[686,687,688,689],[583,1781,612],{"className":1782},[652,656,689],[583,1784,707],{"className":1785},[706],[583,1787,1789],{"className":1788},[669],[583,1790,1792],{"className":1791,"style":775},[673],[583,1793],{},[583,1795,1797,1800],{"style":1796},"top:-2.7922em;",[583,1798],{"className":1799,"style":1181},[681],[583,1801,1803],{"className":1802,"style":1199},[1198],[1201,1804,1805],{"xmlns":1203,"width":1204,"height":1205,"viewBox":1206,"preserveAspectRatio":1207},[1209,1806],{"d":1211},[583,1808,707],{"className":1809},[706],[583,1811,1813],{"className":1812},[669],[583,1814,1817],{"className":1815,"style":1816},[673],"height:0.2078em;",[583,1818],{},": Die ersten Glieder sind ",[583,1821,1823,1873],{"className":1822},[586],[583,1824,1826],{"className":1825},[590],[592,1827,1828],{"xmlns":594},[596,1829,1830,1870],{},[599,1831,1832,1834,1836,1838,1843,1846,1849,1851,1853,1865,1867],{},[618,1833,620],{},[614,1835,1029],{"separator":639},[1031,1837,1033],{},[1141,1839,1840],{},[618,1841,1842],{},"3",[614,1844,1845],{},"≈",[618,1847,1848],{},"1,7321",[614,1850,1029],{"separator":639},[1031,1852,1033],{},[1141,1854,1855],{},[599,1856,1857,1859,1861],{},[618,1858,1145],{},[614,1860,616],{},[1141,1862,1863],{},[618,1864,1842],{},[614,1866,1845],{},[618,1868,1869],{},"1,9319",[631,1871,1872],{"encoding":633},"1;\\ \\sqrt{3} \\approx 1{,}7321;\\ \\sqrt{2 + \\sqrt{3}} \\approx 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Folge konvergiert also. Der Grenzwert erfüllt ",[583,2824,2826,2852],{"className":2825},[586],[583,2827,2829],{"className":2828},[590],[592,2830,2831],{"xmlns":594},[596,2832,2833,2849],{},[599,2834,2835,2837,2839],{},[605,2836,607],{},[614,2838,1258],{},[1141,2840,2841],{},[599,2842,2843,2845,2847],{},[618,2844,1145],{},[614,2846,616],{},[605,2848,607],{},[631,2850,2851],{"encoding":633},"a = \\sqrt{2 + a}",[583,2853,2855,2873],{"className":2854,"ariaHidden":639},[638],[583,2856,2858,2861,2864,2867,2870],{"className":2857},[643],[583,2859],{"className":2860,"style":804},[647],[583,2862,607],{"className":2863},[652,656],[583,2865],{"className":2866,"style":721},[720],[583,2868,1258],{"className":2869},[725],[583,2871],{"className":2872,"style":721},[720],[583,2874,2876,2879],{"className":2875},[643],[583,2877],{"className":2878,"style":2757},[647],[583,2880,2882],{"className":2881},[652,1162],[583,2883,2885,2930],{"className":2884},[664,665],[583,2886,2888,2927],{"className":2887},[669],[583,2889,2891,2915],{"className":2890,"style":2770},[673],[583,2892,2894,2897],{"className":2893,"style":1177},[1176],[583,2895],{"className":2896,"style":1181},[681],[583,2898,2900,2903,2906,2909,2912],{"className":2899,"style":1185},[652],[583,2901,1145],{"className":2902},[652],[583,2904],{"className":2905,"style":1747},[720],[583,2907,616],{"className":2908},[699],[583,2910],{"className":2911,"style":1747},[720],[583,2913,607],{"className":2914},[652,656],[583,2916,2917,2920],{"style":2797},[583,2918],{"className":2919,"style":1181},[681],[583,2921,2923],{"className":2922,"style":1199},[1198],[1201,2924,2925],{"xmlns":1203,"width":1204,"height":1205,"viewBox":1206,"preserveAspectRatio":1207},[1209,2926],{"d":1211},[583,2928,707],{"className":2929},[706],[583,2931,2933],{"className":2932},[669],[583,2934,2936],{"className":2935,"style":2817},[673],[583,2937],{},", d.h. ",[583,2940,2942,2972],{"className":2941},[586],[583,2943,2945],{"className":2944},[590],[592,2946,2947],{"xmlns":594},[596,2948,2949,2969],{},[599,2950,2951,2957,2959,2961,2963,2965,2967],{},[2491,2952,2953,2955],{},[605,2954,607],{},[618,2956,1145],{},[614,2958,2499],{},[605,2960,607],{},[614,2962,2499],{},[618,2964,1145],{},[614,2966,1258],{},[618,2968,2415],{},[631,2970,2971],{"encoding":633},"a^2 - a - 2 = 0",[583,2973,2975,3019,3037,3055],{"className":2974,"ariaHidden":639},[638],[583,2976,2978,2981,3010,3013,3016],{"className":2977},[643],[583,2979],{"className":2980,"style":2544},[647],[583,2982,2984,2987],{"className":2983},[652],[583,2985,607],{"className":2986},[652,656],[583,2988,2990],{"className":2989},[660],[583,2991,2993],{"className":2992},[664],[583,2994,2996],{"className":2995},[669],[583,2997,2999],{"className":2998,"style":2563},[673],[583,3000,3001,3004],{"style":2566},[583,3002],{"className":3003,"style":682},[681],[583,3005,3007],{"className":3006},[686,687,688,689],[583,3008,1145],{"className":3009},[652,689],[583,3011],{"className":3012,"style":1747},[720],[583,3014,2499],{"className":3015},[699],[583,3017],{"className":3018,"style":1747},[720],[583,3020,3022,3025,3028,3031,3034],{"className":3021},[643],[583,3023],{"className":3024,"style":2591},[647],[583,3026,607],{"className":3027},[652,656],[583,3029],{"className":3030,"style":1747},[720],[583,3032,2499],{"className":3033},[699],[583,3035],{"className":3036,"style":1747},[720],[583,3038,3040,3043,3046,3049,3052],{"className":3039},[643],[583,3041],{"className":3042,"style":1597},[647],[583,3044,1145],{"className":3045},[652],[583,3047],{"className":3048,"style":721},[720],[583,3050,1258],{"className":3051},[725],[583,3053],{"className":3054,"style":721},[720],[583,3056,3058,3061],{"className":3057},[643],[583,3059],{"className":3060,"style":1597},[647],[583,3062,2415],{"className":3063},[652]," mit den Lösungen ",[583,3066,3068,3081],{"className":3067},[586],[583,3069,3071],{"className":3070},[590],[592,3072,3073],{"xmlns":594},[596,3074,3075,3079],{},[599,3076,3077],{},[618,3078,1145],{},[631,3080,1145],{"encoding":633},[583,3082,3084],{"className":3083,"ariaHidden":639},[638],[583,3085,3087,3090],{"className":3086},[643],[583,3088],{"className":3089,"style":1597},[647],[583,3091,1145],{"className":3092},[652]," und ",[583,3095,3097,3113],{"className":3096},[586],[583,3098,3100],{"className":3099},[590],[592,3101,3102],{"xmlns":594},[596,3103,3104,3110],{},[599,3105,3106,3108],{},[614,3107,2499],{},[618,3109,620],{},[631,3111,3112],{"encoding":633},"-1",[583,3114,3116],{"className":3115,"ariaHidden":639},[638],[583,3117,3119,3123,3126],{"className":3118},[643],[583,3120],{"className":3121,"style":3122},[647],"height:0.7278em;vertical-align:-0.0833em;",[583,3124,2499],{"className":3125},[652],[583,3127,620],{"className":3128},[652],". Da alle Glieder positiv sind, ist ",[583,3131,3133,3151],{"className":3132},[586],[583,3134,3136],{"className":3135},[590],[592,3137,3138],{"xmlns":594},[596,3139,3140,3148],{},[599,3141,3142,3144,3146],{},[605,3143,607],{},[614,3145,1258],{},[618,3147,1145],{},[631,3149,3150],{"encoding":633},"a = 2",[583,3152,3154,3172],{"className":3153,"ariaHidden":639},[638],[583,3155,3157,3160,3163,3166,3169],{"className":3156},[643],[583,3158],{"className":3159,"style":804},[647],[583,3161,607],{"className":3162},[652,656],[583,3164],{"className":3165,"style":721},[720],[583,3167,1258],{"className":3168},[725],[583,3170],{"className":3171,"style":721},[720],[583,3173,3175,3178],{"className":3174},[643],[583,3176],{"className":3177,"style":1597},[647],[583,3179,1145],{"className":3180},[652],[564,3182,3183,3427,3428,3504,3505,3533,3534,3563,3564,3616,3617,2121],{},[583,3184,3186,3227],{"className":3185},[586],[583,3187,3189],{"className":3188},[590],[592,3190,3191],{"xmlns":594},[596,3192,3193,3224],{},[599,3194,3195,3201,3203],{},[602,3196,3197,3199],{},[605,3198,607],{},[605,3200,612],{},[614,3202,1258],{},[2491,3204,3205,3222],{},[599,3206,3207,3209,3211,3213,3220],{},[614,3208,1265],{"fence":639},[618,3210,620],{},[614,3212,616],{},[3214,3215,3216,3218],"mfrac",{},[618,3217,620],{},[605,3219,612],{},[614,3221,1274],{"fence":639},[605,3223,612],{},[631,3225,3226],{"encoding":633},"a_n = \\left(1 + \\frac{1}{n}\\right)^n",[583,3228,3230,3285],{"className":3229,"ariaHidden":639},[638],[583,3231,3233,3236,3276,3279,3282],{"className":3232},[643],[583,3234],{"className":3235,"style":735},[647],[583,3237,3239,3242],{"className":3238},[652],[583,3240,607],{"className":3241},[652,656],[583,3243,3245],{"className":3244},[660],[583,3246,3248,3268],{"className":3247},[664,665],[583,3249,3251,3265],{"className":3250},[669],[583,3252,3254],{"className":3253,"style":754},[673],[583,3255,3256,3259],{"style":677},[583,3257],{"className":3258,"style":682},[681],[583,3260,3262],{"className":3261},[686,687,688,689],[583,3263,612],{"className":3264},[652,656,689],[583,3266,707],{"className":3267},[706],[583,3269,3271],{"className":3270},[669],[583,3272,3274],{"className":3273,"style":775},[673],[583,3275],{},[583,3277],{"className":3278,"style":721},[720],[583,3280,1258],{"className":3281},[725],[583,3283],{"className":3284,"style":721},[720],[583,3286,3288,3292],{"className":3287},[643],[583,3289],{"className":3290,"style":3291},[647],"height:1.2543em;vertical-align:-0.35em;",[583,3293,3295,3402],{"className":3294},[1126],[583,3296,3298,3308,3311,3314,3317,3320,3396],{"className":3297},[1126],[583,3299,3303],{"className":3300,"style":3302},[1360,3301],"delimcenter","top:0em;",[583,3304,1265],{"className":3305},[3306,3307],"delimsizing","size1",[583,3309,620],{"className":3310},[652],[583,3312],{"className":3313,"style":1747},[720],[583,3315,616],{"className":3316},[699],[583,3318],{"className":3319,"style":1747},[720],[583,3321,3323,3327,3393],{"className":3322},[652],[583,3324],{"className":3325},[1360,3326],"nulldelimiter",[583,3328,3330],{"className":3329},[3214],[583,3331,3333,3384],{"className":3332},[664,665],[583,3334,3336,3381],{"className":3335},[669],[583,3337,3340,3355,3366],{"className":3338,"style":3339},[673],"height:0.8451em;",[583,3341,3343,3346],{"style":3342},"top:-2.655em;",[583,3344],{"className":3345,"style":1181},[681],[583,3347,3349],{"className":3348},[686,687,688,689],[583,3350,3352],{"className":3351},[652,689],[583,3353,612],{"className":3354},[652,656,689],[583,3356,3358,3361],{"style":3357},"top:-3.23em;",[583,3359],{"className":3360,"style":1181},[681],[583,3362],{"className":3363,"style":3365},[3364],"frac-line","border-bottom-width:0.04em;",[583,3367,3369,3372],{"style":3368},"top:-3.394em;",[583,3370],{"className":3371,"style":1181},[681],[583,3373,3375],{"className":3374},[686,687,688,689],[583,3376,3378],{"className":3377},[652,689],[583,3379,620],{"className":3380},[652,689],[583,3382,707],{"className":3383},[706],[583,3385,3387],{"className":3386},[669],[583,3388,3391],{"className":3389,"style":3390},[673],"height:0.345em;",[583,3392],{},[583,3394],{"className":3395},[1404,3326],[583,3397,3399],{"className":3398,"style":3302},[1404,3301],[583,3400,1274],{"className":3401},[3306,3307],[583,3403,3405],{"className":3404},[660],[583,3406,3408],{"className":3407},[664],[583,3409,3411],{"className":3410},[669],[583,3412,3415],{"className":3413,"style":3414},[673],"height:0.9043em;",[583,3416,3418,3421],{"style":3417},"top:-3.3029em;margin-right:0.05em;",[583,3419],{"className":3420,"style":682},[681],[583,3422,3424],{"className":3423},[686,687,688,689],[583,3425,612],{"className":3426},[652,656,689]," ist wachsend (",[583,3429,3431,3458],{"className":3430},[586],[583,3432,3434],{"className":3433},[590],[592,3435,3436],{"xmlns":594},[596,3437,3438,3455],{},[599,3439,3440,3442,3444,3446,3449,3451,3453],{},[618,3441,1145],{},[614,3443,1029],{"separator":639},[1031,3445,1033],{},[618,3447,3448],{},"2,25",[614,3450,1029],{"separator":639},[1031,3452,1033],{},[614,3454,1050],{},[631,3456,3457],{"encoding":633},"2;\\ 2{,}25;\\ \\ldots",[583,3459,3461],{"className":3460,"ariaHidden":639},[638],[583,3462,3464,3467,3470,3473,3476,3479,3482,3488,3492,3495,3498,3501],{"className":3463},[643],[583,3465],{"className":3466,"style":1063},[647],[583,3468,1145],{"className":3469},[652],[583,3471,1029],{"className":3472},[1070],[583,3474,1033],{"className":3475},[720],[583,3477],{"className":3478,"style":1077},[720],[583,3480,1145],{"className":3481},[652],[583,3483,3485],{"className":3484},[652],[583,3486,1087],{"className":3487},[1070],[583,3489,3491],{"className":3490},[652],"25",[583,3493,1029],{"className":3494},[1070],[583,3496,1033],{"className":3497},[720],[583,3499],{"className":3500,"style":1077},[720],[583,3502,1050],{"className":3503},[1126],") und durch ",[583,3506,3508,3521],{"className":3507},[586],[583,3509,3511],{"className":3510},[590],[592,3512,3513],{"xmlns":594},[596,3514,3515,3519],{},[599,3516,3517],{},[618,3518,1842],{},[631,3520,1842],{"encoding":633},[583,3522,3524],{"className":3523,"ariaHidden":639},[638],[583,3525,3527,3530],{"className":3526},[643],[583,3528],{"className":3529,"style":1597},[647],[583,3531,1842],{"className":3532},[652]," nach oben beschränkt. Ihr Grenzwert ist die Eulersche Zahl ",[583,3535,3537,3551],{"className":3536},[586],[583,3538,3540],{"className":3539},[590],[592,3541,3542],{"xmlns":594},[596,3543,3544,3549],{},[599,3545,3546],{},[605,3547,3548],{},"e",[631,3550,3548],{"encoding":633},[583,3552,3554],{"className":3553,"ariaHidden":639},[638],[583,3555,3557,3560],{"className":3556},[643],[583,3558],{"className":3559,"style":804},[647],[583,3561,3548],{"className":3562},[652,656],". Für ",[583,3565,3567,3586],{"className":3566},[586],[583,3568,3570],{"className":3569},[590],[592,3571,3572],{"xmlns":594},[596,3573,3574,3583],{},[599,3575,3576,3578,3580],{},[605,3577,612],{},[614,3579,1258],{},[618,3581,3582],{},"1000",[631,3584,3585],{"encoding":633},"n = 1000",[583,3587,3589,3607],{"className":3588,"ariaHidden":639},[638],[583,3590,3592,3595,3598,3601,3604],{"className":3591},[643],[583,3593],{"className":3594,"style":804},[647],[583,3596,612],{"className":3597},[652,656],[583,3599],{"className":3600,"style":721},[720],[583,3602,1258],{"className":3603},[725],[583,3605],{"className":3606,"style":721},[720],[583,3608,3610,3613],{"className":3609},[643],[583,3611],{"className":3612,"style":1597},[647],[583,3614,3582],{"className":3615},[652]," ist ",[583,3618,3620,3643],{"className":3619},[586],[583,3621,3623],{"className":3622},[590],[592,3624,3625],{"xmlns":594},[596,3626,3627,3640],{},[599,3628,3629,3635,3637],{},[602,3630,3631,3633],{},[605,3632,607],{},[605,3634,612],{},[614,3636,1845],{},[618,3638,3639],{},"2,7169",[631,3641,3642],{"encoding":633},"a_n \\approx 2{,}7169",[583,3644,3646,3702],{"className":3645,"ariaHidden":639},[638],[583,3647,3649,3653,3693,3696,3699],{"className":3648},[643],[583,3650],{"className":3651,"style":3652},[647],"height:0.6331em;vertical-align:-0.15em;",[583,3654,3656,3659],{"className":3655},[652],[583,3657,607],{"className":3658},[652,656],[583,3660,3662],{"className":3661},[660],[583,3663,3665,3685],{"className":3664},[664,665],[583,3666,3668,3682],{"className":3667},[669],[583,3669,3671],{"className":3670,"style":754},[673],[583,3672,3673,3676],{"style":677},[583,3674],{"className":3675,"style":682},[681],[583,3677,3679],{"className":3678},[686,687,688,689],[583,3680,612],{"className":3681},[652,656,689],[583,3683,707],{"className":3684},[706],[583,3686,3688],{"className":3687},[669],[583,3689,3691],{"className":3690,"style":775},[673],[583,3692],{},[583,3694],{"className":3695,"style":721},[720],[583,3697,1845],{"className":3698},[725],[583,3700],{"className":3701,"style":721},[720],[583,3703,3705,3708,3711,3717],{"className":3704},[643],[583,3706],{"className":3707,"style":1063},[647],[583,3709,1145],{"className":3710},[652],[583,3712,3714],{"className":3713},[652],[583,3715,1087],{"className":3716},[1070],[583,3718,3720],{"className":3719},[652],"7169",[559,3722,3724],{"id":3723},"typische-fehler","Typische Fehler",[573,3726,3727,3806,4233,4494],{},[576,3728,3729,3730,3805],{},"Beschränktheit allein für ausreichend halten. ",[583,3731,3733,3757],{"className":3732},[586],[583,3734,3736],{"className":3735},[590],[592,3737,3738],{"xmlns":594},[596,3739,3740,3754],{},[599,3741,3742,3744,3746,3748],{},[614,3743,1265],{"stretchy":1264},[614,3745,2499],{},[618,3747,620],{},[2491,3749,3750,3752],{},[614,3751,1274],{"stretchy":1264},[605,3753,612],{},[631,3755,3756],{"encoding":633},"(-1)^n",[583,3758,3760],{"className":3759,"ariaHidden":639},[638],[583,3761,3763,3766,3769,3772,3775],{"className":3762},[643],[583,3764],{"className":3765,"style":1352},[647],[583,3767,1265],{"className":3768},[1360],[583,3770,2499],{"className":3771},[652],[583,3773,620],{"className":3774},[652],[583,3776,3778,3781],{"className":3777},[1404],[583,3779,1274],{"className":3780},[1404],[583,3782,3784],{"className":3783},[660],[583,3785,3787],{"className":3786},[664],[583,3788,3790],{"className":3789},[669],[583,3791,3794],{"className":3792,"style":3793},[673],"height:0.6644em;",[583,3795,3796,3799],{"style":2566},[583,3797],{"className":3798,"style":682},[681],[583,3800,3802],{"className":3801},[686,687,688,689],[583,3803,612],{"className":3804},[652,656,689]," ist beschränkt, aber nicht monoton, und divergiert.",[576,3807,3808,3809,3963,3964,3616,4055,4106,4107,4163,4164,4232],{},"Den Grenzwert aus der Fixpunktgleichung berechnen, ohne die Konvergenz zu zeigen. Bei ",[583,3810,3812,3846],{"className":3811},[586],[583,3813,3815],{"className":3814},[590],[592,3816,3817],{"xmlns":594},[596,3818,3819,3843],{},[599,3820,3821,3833,3835,3837],{},[602,3822,3823,3825],{},[605,3824,607],{},[599,3826,3827,3829,3831],{},[605,3828,612],{},[614,3830,616],{},[618,3832,620],{},[614,3834,1258],{},[618,3836,1145],{},[602,3838,3839,3841],{},[605,3840,607],{},[605,3842,612],{},[631,3844,3845],{"encoding":633},"a_{n+1} = 2a_n",[583,3847,3849,3913],{"className":3848,"ariaHidden":639},[638],[583,3850,3852,3855,3904,3907,3910],{"className":3851},[643],[583,3853],{"className":3854,"style":1287},[647],[583,3856,3858,3861],{"className":3857},[652],[583,3859,607],{"className":3860},[652,656],[583,3862,3864],{"className":3863},[660],[583,3865,3867,3896],{"className":3866},[664,665],[583,3868,3870,3893],{"className":3869},[669],[583,3871,3873],{"className":3872,"style":674},[673],[583,3874,3875,3878],{"style":677},[583,3876],{"className":3877,"style":682},[681],[583,3879,3881],{"className":3880},[686,687,688,689],[583,3882,3884,3887,3890],{"className":3883},[652,689],[583,3885,612],{"className":3886},[652,656,689],[583,3888,616],{"className":3889},[699,689],[583,3891,620],{"className":3892},[652,689],[583,3894,707],{"className":3895},[706],[583,3897,3899],{"className":3898},[669],[583,3900,3902],{"className":3901,"style":714},[673],[583,3903],{},[583,3905],{"className":3906,"style":721},[720],[583,3908,1258],{"className":3909},[725],[583,3911],{"className":3912,"style":721},[720],[583,3914,3916,3920,3923],{"className":3915},[643],[583,3917],{"className":3918,"style":3919},[647],"height:0.7944em;vertical-align:-0.15em;",[583,3921,1145],{"className":3922},[652],[583,3924,3926,3929],{"className":3925},[652],[583,3927,607],{"className":3928},[652,656],[583,3930,3932],{"className":3931},[660],[583,3933,3935,3955],{"className":3934},[664,665],[583,3936,3938,3952],{"className":3937},[669],[583,3939,3941],{"className":3940,"style":754},[673],[583,3942,3943,3946],{"style":677},[583,3944],{"className":3945,"style":682},[681],[583,3947,3949],{"className":3948},[686,687,688,689],[583,3950,612],{"className":3951},[652,656,689],[583,3953,707],{"className":3954},[706],[583,3956,3958],{"className":3957},[669],[583,3959,3961],{"className":3960,"style":775},[673],[583,3962],{}," mit ",[583,3965,3967,3988],{"className":3966},[586],[583,3968,3970],{"className":3969},[590],[592,3971,3972],{"xmlns":594},[596,3973,3974,3986],{},[599,3975,3976,3982,3984],{},[602,3977,3978,3980],{},[605,3979,607],{},[618,3981,620],{},[614,3983,1258],{},[618,3985,620],{},[631,3987,1532],{"encoding":633},[583,3989,3991,4046],{"className":3990,"ariaHidden":639},[638],[583,3992,3994,3997,4037,4040,4043],{"className":3993},[643],[583,3995],{"className":3996,"style":735},[647],[583,3998,4000,4003],{"className":3999},[652],[583,4001,607],{"className":4002},[652,656],[583,4004,4006],{"className":4005},[660],[583,4007,4009,4029],{"className":4008},[664,665],[583,4010,4012,4026],{"className":4011},[669],[583,4013,4015],{"className":4014,"style":674},[673],[583,4016,4017,4020],{"style":677},[583,4018],{"className":4019,"style":682},[681],[583,4021,4023],{"className":4022},[686,687,688,689],[583,4024,620],{"className":4025},[652,689],[583,4027,707],{"className":4028},[706],[583,4030,4032],{"className":4031},[669],[583,4033,4035],{"className":4034,"style":775},[673],[583,4036],{},[583,4038],{"className":4039,"style":721},[720],[583,4041,1258],{"className":4042},[725],[583,4044],{"className":4045,"style":721},[720],[583,4047,4049,4052],{"className":4048},[643],[583,4050],{"className":4051,"style":1597},[647],[583,4053,620],{"className":4054},[652],[583,4056,4058,4076],{"className":4057},[586],[583,4059,4061],{"className":4060},[590],[592,4062,4063],{"xmlns":594},[596,4064,4065,4073],{},[599,4066,4067,4069,4071],{},[605,4068,607],{},[614,4070,1258],{},[618,4072,2415],{},[631,4074,4075],{"encoding":633},"a = 0",[583,4077,4079,4097],{"className":4078,"ariaHidden":639},[638],[583,4080,4082,4085,4088,4091,4094],{"className":4081},[643],[583,4083],{"className":4084,"style":804},[647],[583,4086,607],{"className":4087},[652,656],[583,4089],{"className":4090,"style":721},[720],[583,4092,1258],{"className":4093},[725],[583,4095],{"className":4096,"style":721},[720],[583,4098,4100,4103],{"className":4099},[643],[583,4101],{"className":4102,"style":1597},[647],[583,4104,2415],{"className":4105},[652]," die einzige Lösung von ",[583,4108,4110,4130],{"className":4109},[586],[583,4111,4113],{"className":4112},[590],[592,4114,4115],{"xmlns":594},[596,4116,4117,4127],{},[599,4118,4119,4121,4123,4125],{},[605,4120,607],{},[614,4122,1258],{},[618,4124,1145],{},[605,4126,607],{},[631,4128,4129],{"encoding":633},"a = 2a",[583,4131,4133,4151],{"className":4132,"ariaHidden":639},[638],[583,4134,4136,4139,4142,4145,4148],{"className":4135},[643],[583,4137],{"className":4138,"style":804},[647],[583,4140,607],{"className":4141},[652,656],[583,4143],{"className":4144,"style":721},[720],[583,4146,1258],{"className":4147},[725],[583,4149],{"className":4150,"style":721},[720],[583,4152,4154,4157,4160],{"className":4153},[643],[583,4155],{"className":4156,"style":1597},[647],[583,4158,1145],{"className":4159},[652],[583,4161,607],{"className":4162},[652,656],", die Folge ",[583,4165,4167,4193],{"className":4166},[586],[583,4168,4170],{"className":4169},[590],[592,4171,4172],{"xmlns":594},[596,4173,4174,4190],{},[599,4175,4176,4178,4180,4182,4184,4186,4188],{},[618,4177,620],{},[614,4179,1087],{"separator":639},[618,4181,1145],{},[614,4183,1087],{"separator":639},[618,4185,1091],{},[614,4187,1087],{"separator":639},[614,4189,1050],{},[631,4191,4192],{"encoding":633},"1, 2, 4, \\ldots",[583,4194,4196],{"className":4195,"ariaHidden":639},[638],[583,4197,4199,4202,4205,4208,4211,4214,4217,4220,4223,4226,4229],{"className":4198},[643],[583,4200],{"className":4201,"style":1063},[647],[583,4203,620],{"className":4204},[652],[583,4206,1087],{"className":4207},[1070],[583,4209],{"className":4210,"style":1077},[720],[583,4212,1145],{"className":4213},[652],[583,4215,1087],{"className":4216},[1070],[583,4218],{"className":4219,"style":1077},[720],[583,4221,1091],{"className":4222},[652],[583,4224,1087],{"className":4225},[1070],[583,4227],{"className":4228,"style":1077},[720],[583,4230,1050],{"className":4231},[1126]," divergiert aber.",[576,4234,4235,4236,4401,4402,4430,4431,2121],{},"Die Schranke für den Grenzwert halten. ",[583,4237,4239,4271],{"className":4238},[586],[583,4240,4242],{"className":4241},[590],[592,4243,4244],{"xmlns":594},[596,4245,4246,4268],{},[599,4247,4248],{},[2491,4249,4250,4266],{},[599,4251,4252,4254,4256,4258,4264],{},[614,4253,1265],{"fence":639},[618,4255,620],{},[614,4257,616],{},[3214,4259,4260,4262],{},[618,4261,620],{},[605,4263,612],{},[614,4265,1274],{"fence":639},[605,4267,612],{},[631,4269,4270],{"encoding":633},"\\left(1 + \\frac{1}{n}\\right)^n",[583,4272,4274],{"className":4273,"ariaHidden":639},[638],[583,4275,4277,4280],{"className":4276},[643],[583,4278],{"className":4279,"style":3291},[647],[583,4281,4283,4378],{"className":4282},[1126],[583,4284,4286,4292,4295,4298,4301,4304,4372],{"className":4285},[1126],[583,4287,4289],{"className":4288,"style":3302},[1360,3301],[583,4290,1265],{"className":4291},[3306,3307],[583,4293,620],{"className":4294},[652],[583,4296],{"className":4297,"style":1747},[720],[583,4299,616],{"className":4300},[699],[583,4302],{"className":4303,"style":1747},[720],[583,4305,4307,4310,4369],{"className":4306},[652],[583,4308],{"className":4309},[1360,3326],[583,4311,4313],{"className":4312},[3214],[583,4314,4316,4361],{"className":4315},[664,665],[583,4317,4319,4358],{"className":4318},[669],[583,4320,4322,4336,4344],{"className":4321,"style":3339},[673],[583,4323,4324,4327],{"style":3342},[583,4325],{"className":4326,"style":1181},[681],[583,4328,4330],{"className":4329},[686,687,688,689],[583,4331,4333],{"className":4332},[652,689],[583,4334,612],{"className":4335},[652,656,689],[583,4337,4338,4341],{"style":3357},[583,4339],{"className":4340,"style":1181},[681],[583,4342],{"className":4343,"style":3365},[3364],[583,4345,4346,4349],{"style":3368},[583,4347],{"className":4348,"style":1181},[681],[583,4350,4352],{"className":4351},[686,687,688,689],[583,4353,4355],{"className":4354},[652,689],[583,4356,620],{"className":4357},[652,689],[583,4359,707],{"className":4360},[706],[583,4362,4364],{"className":4363},[669],[583,4365,4367],{"className":4366,"style":3390},[673],[583,4368],{},[583,4370],{"className":4371},[1404,3326],[583,4373,4375],{"className":4374,"style":3302},[1404,3301],[583,4376,1274],{"className":4377},[3306,3307],[583,4379,4381],{"className":4380},[660],[583,4382,4384],{"className":4383},[664],[583,4385,4387],{"className":4386},[669],[583,4388,4390],{"className":4389,"style":3414},[673],[583,4391,4392,4395],{"style":3417},[583,4393],{"className":4394,"style":682},[681],[583,4396,4398],{"className":4397},[686,687,688,689],[583,4399,612],{"className":4400},[652,656,689]," ist durch ",[583,4403,4405,4418],{"className":4404},[586],[583,4406,4408],{"className":4407},[590],[592,4409,4410],{"xmlns":594},[596,4411,4412,4416],{},[599,4413,4414],{},[618,4415,1842],{},[631,4417,1842],{"encoding":633},[583,4419,4421],{"className":4420,"ariaHidden":639},[638],[583,4422,4424,4427],{"className":4423},[643],[583,4425],{"className":4426,"style":1597},[647],[583,4428,1842],{"className":4429},[652]," beschränkt, der Grenzwert ist ",[583,4432,4434,4453],{"className":4433},[586],[583,4435,4437],{"className":4436},[590],[592,4438,4439],{"xmlns":594},[596,4440,4441,4450],{},[599,4442,4443,4445,4447],{},[605,4444,3548],{},[614,4446,1845],{},[618,4448,4449],{},"2,718",[631,4451,4452],{"encoding":633},"e \\approx 2{,}718",[583,4454,4456,4475],{"className":4455,"ariaHidden":639},[638],[583,4457,4459,4463,4466,4469,4472],{"className":4458},[643],[583,4460],{"className":4461,"style":4462},[647],"height:0.4831em;",[583,4464,3548],{"className":4465},[652,656],[583,4467],{"className":4468,"style":721},[720],[583,4470,1845],{"className":4471},[725],[583,4473],{"className":4474,"style":721},[720],[583,4476,4478,4481,4484,4490],{"className":4477},[643],[583,4479],{"className":4480,"style":1063},[647],[583,4482,1145],{"className":4483},[652],[583,4485,4487],{"className":4486},[652],[583,4488,1087],{"className":4489},[1070],[583,4491,4493],{"className":4492},[652],"718",[576,4495,4496],{},"Bei wachsenden Folgen eine untere Schranke angeben. Für wachsende Folgen ist die obere Schranke entscheidend.",{"title":4498,"searchDepth":4499,"depth":4499,"links":4500},"",2,[4501,4502,4503],{"id":561,"depth":4499,"text":562},{"id":1504,"depth":4499,"text":1505},{"id":3723,"depth":4499,"text":3724},[4505,4508,4511,4513,4515,4517,4519,4522,4525],{"lhs":4506,"rhs":4507},"round(sqrt(2 + 1), 4)","1.7321",{"lhs":4509,"rhs":4510},"round(sqrt(2 + sqrt(2 + 1)), 4)","1.9319",{"lhs":4512,"rhs":1145},"sqrt(2 + 2)",{"lhs":4514,"rhs":2415},"2^2 - 2 - 2",{"lhs":4516,"rhs":2415},"(-1)^2 - (-1) - 2",{"lhs":4518,"rhs":1145},"(1 + 1\u002F1)^1",{"lhs":4520,"rhs":4521},"(1 + 1\u002F2)^2","2.25",{"lhs":4523,"rhs":4524},"round((1 + 1\u002F1000)^1000, 4)","2.7169",{"lhs":4526,"rhs":2415},"2*0","(a_n) \\text{ reelle Folge}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(a_n)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">reelle Folge\u003C\u002Fspan>",null,"md","monotone-convergence","(a_n) \\text{ monoton wachsend und nach oben beschränkt} \\;\\Rightarrow\\; \\lim_{n \\to \\infty} a_n = \\sup_{n \\in \\mathbb{N}} a_n","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmrow>\u003Cmtext> monoton wachsend und nach oben beschr\u003C\u002Fmtext>\u003Cmover accent=\"true\">\u003Cmtext>a\u003C\u002Fmtext>\u003Cmo>¨\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmtext>nkt\u003C\u002Fmtext>\u003C\u002Fmrow>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmo>⇒\u003C\u002Fmo>\u003Cmtext>  \u003C\u002Fmtext>\u003Cmunder>\u003Cmrow>\u003Cmi>lim\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo>=\u003C\u002Fmo>\u003Cmunder>\u003Cmrow>\u003Cmi>sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>∈\u003C\u002Fmo>\u003Cmi mathvariant=\"double-struck\">N\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(a_n) \\text{ monoton wachsend und nach oben beschränkt} \\;\\Rightarrow\\; \\lim_{n \\to \\infty} a_n = \\sup_{n \\in \\mathbb{N}} a_n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mord text\">\u003Cspan class=\"mord\"> monoton wachsend und nach oben beschr\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6679em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">a\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.25em;\">\u003Cspan class=\"mord\">¨\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mord\">nkt\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">⇒\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.3944em;vertical-align:-0.7em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6944em;\">\u003Cspan style=\"top:-2.4em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">→\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">lim\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.7em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.4006em;vertical-align:-0.97em;\">\u003C\u002Fspan>\u003Cspan class=\"mop op-limits\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.4306em;\">\u003Cspan style=\"top:-2.1573em;margin-left:0em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">∈\u003C\u002Fspan>\u003Cspan class=\"mord mathbb mtight\">N\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan>\u003Cspan class=\"mop\">sup\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.97em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","university",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Fsequences-series\u002Fmonotoniekriterium-fuer-folgen",[],[4540,4541,4542],"limit-definition","euler-number-limit","fixed-point-iteration",{"title":548,"description":4498},"de\u002Fformulas\u002Fmath\u002Fsequences-series\u002Fmonotoniekriterium-fuer-folgen","Jede monoton wachsende, nach oben beschränkte Folge konvergiert gegen ihr Supremum; ebenso fallend und nach unten beschränkt.",[4547,4548,4549,4550],"folge","konvergenz","monotonie","beschränktheit",[4552,4556,4560],{"symbol":4553,"meaning":4554,"symbolHtml":4555},"a_n","Glieder der Folge","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a_n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5806em;vertical-align:-0.15em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4557,"meaning":4558,"symbolHtml":4559},"\\sup a_n","Supremum, die kleinste obere Schranke der Folgenglieder","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sup a_n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">sup\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">a\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.1514em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":612,"meaning":4561,"symbolHtml":4562},"Index der Folgenglieder","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">n\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","ne70P34RirNNLkvjqkdpQhxoTrDYbI_5genhQNs96Ec",[4565,4568,4571],{"stem":4566,"title":4567,"category":101},"de\u002Fformulas\u002Fmath\u002Fcalculus\u002Fgrenzwert-einer-folge","Grenzwert einer Folge (ε-Definition)",{"stem":4569,"title":4570,"category":391},"de\u002Fformulas\u002Fmath\u002Fnumerics\u002Ffixpunktiteration","Fixpunktiteration",{"stem":4572,"title":4573,"category":412},"de\u002Fformulas\u002Fmath\u002Fsequences-series\u002Feulersche-zahl-grenzwert","Eulersche Zahl als Grenzwert",1793824264482]