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x_0| > r",[573,1131,1133,1154,1212],{"className":1132,"ariaHidden":640},[639],[573,1134,1136,1139,1142,1145,1148,1151],{"className":1135},[644],[573,1137],{"className":1138,"style":649},[648],[573,1140,975],{"className":1141},[665],[573,1143,611],{"className":1144},[665,669],[573,1146],{"className":1147,"style":730},[660],[573,1149,614],{"className":1150},[734],[573,1152],{"className":1153,"style":730},[660],[573,1155,1157,1160,1200,1203,1206,1209],{"className":1156},[644],[573,1158],{"className":1159,"style":649},[648],[573,1161,1163,1166],{"className":1162},[665],[573,1164,611],{"className":1165},[665,669],[573,1167,1169],{"className":1168},[673],[573,1170,1172,1192],{"className":1171},[677,678],[573,1173,1175,1189],{"className":1174},[682],[573,1176,1178],{"className":1177,"style":762},[686],[573,1179,1180,1183],{"style":690},[573,1181],{"className":1182,"style":695},[694],[573,1184,1186],{"className":1185},[699,700,701,702],[573,1187,622],{"className":1188},[665,702],[573,1190,710],{"className":1191},[709],[573,1193,1195],{"className":1194},[682],[573,1196,1198],{"className":1197,"style":717},[686],[573,1199],{},[573,1201,975],{"className":1202},[665],[573,1204],{"className":1205,"style":1072},[660],[573,1207,1124],{"className":1208},[1076],[573,1210],{"className":1211,"style":1072},[660],[573,1213,1215,1218],{"className":1214},[644],[573,1216],{"className":1217,"style":843},[648],[573,1219,937],{"className":1220,"style":952},[665,669],": Die Reihe divergiert.",[958,1223,1224,1349,1350,571,1353,1447],{},[573,1225,1227,1258],{"className":1226},[576],[573,1228,1230],{"className":1229},[580],[582,1231,1232],{"xmlns":584},[586,1233,1234,1255],{},[589,1235,1236,1238,1240,1242,1248,1250,1253],{},[599,1237,975],{"mathvariant":974},[599,1239,611],{},[592,1241,614],{},[596,1243,1244,1246],{},[599,1245,611],{},[620,1247,622],{},[599,1249,975],{"mathvariant":974},[592,1251,1252],{},"=",[599,1254,937],{},[632,1256,1257],{"encoding":634},"|x - x_0| = r",[573,1259,1261,1282,1340],{"className":1260,"ariaHidden":640},[639],[573,1262,1264,1267,1270,1273,1276,1279],{"className":1263},[644],[573,1265],{"className":1266,"style":649},[648],[573,1268,975],{"className":1269},[665],[573,1271,611],{"className":1272},[665,669],[573,1274],{"className":1275,"style":730},[660],[573,1277,614],{"className":1278},[734],[573,1280],{"className":1281,"style":730},[660],[573,1283,1285,1288,1328,1331,1334,1337],{"className":1284},[644],[573,1286],{"className":1287,"style":649},[648],[573,1289,1291,1294],{"className":1290},[665],[573,1292,611],{"className":1293},[665,669],[573,1295,1297],{"className":1296},[673],[573,1298,1300,1320],{"className":1299},[677,678],[573,1301,1303,1317],{"className":1302},[682],[573,1304,1306],{"className":1305,"style":762},[686],[573,1307,1308,1311],{"style":690},[573,1309],{"className":1310,"style":695},[694],[573,1312,1314],{"className":1313},[699,700,701,702],[573,1315,622],{"className":1316},[665,702],[573,1318,710],{"className":1319},[709],[573,1321,1323],{"className":1322},[682],[573,1324,1326],{"className":1325,"style":717},[686],[573,1327],{},[573,1329,975],{"className":1330},[665],[573,1332],{"className":1333,"style":1072},[660],[573,1335,1252],{"className":1336},[1076],[573,1338],{"className":1339,"style":1072},[660],[573,1341,1343,1346],{"className":1342},[644],[573,1344],{"className":1345,"style":843},[648],[573,1347,937],{"className":1348,"style":952},[665,669],": Keine allgemeine Aussage. Die beiden ",[568,1351,1352],{},"Randpunkte",[573,1354,1356,1379],{"className":1355},[576],[573,1357,1359],{"className":1358},[580],[582,1360,1361],{"xmlns":584},[586,1362,1363,1376],{},[589,1364,1365,1371,1374],{},[596,1366,1367,1369],{},[599,1368,611],{},[620,1370,622],{},[592,1372,1373],{},"±",[599,1375,937],{},[632,1377,1378],{"encoding":634},"x_0 \\pm r",[573,1380,1382,1438],{"className":1381,"ariaHidden":640},[639],[573,1383,1385,1389,1429,1432,1435],{"className":1384},[644],[573,1386],{"className":1387,"style":1388},[648],"height:0.7333em;vertical-align:-0.15em;",[573,1390,1392,1395],{"className":1391},[665],[573,1393,611],{"className":1394},[665,669],[573,1396,1398],{"className":1397},[673],[573,1399,1401,1421],{"className":1400},[677,678],[573,1402,1404,1418],{"className":1403},[682],[573,1405,1407],{"className":1406,"style":762},[686],[573,1408,1409,1412],{"style":690},[573,1410],{"className":1411,"style":695},[694],[573,1413,1415],{"className":1414},[699,700,701,702],[573,1416,622],{"className":1417},[665,702],[573,1419,710],{"className":1420},[709],[573,1422,1424],{"className":1423},[682],[573,1425,1427],{"className":1426,"style":717},[686],[573,1428],{},[573,1430],{"className":1431,"style":730},[660],[573,1433,1373],{"className":1434},[734],[573,1436],{"className":1437,"style":730},[660],[573,1439,1441,1444],{"className":1440},[644],[573,1442],{"className":1443,"style":843},[648],[573,1445,937],{"className":1446,"style":952},[665,669]," prüft man einzeln.",[564,1449,1450,1451,1454],{},"Die erste Form (",[568,1452,1453],{},"Cauchy-Hadamard",") gilt immer. Die zweite Form folgt aus dem Quotientenkriterium und ist meist bequemer, funktioniert aber nur, wenn der Grenzwert existiert. Fehlen Koeffizienten, z.B. bei Reihen mit nur geraden Potenzen, wendet man das Quotientenkriterium direkt auf die Reihenglieder an.",[564,1456,1457],{},"Im Inneren des Konvergenzintervalls darf man die Reihe gliedweise ableiten und integrieren; der Konvergenzradius bleibt dabei gleich.",[559,1459,1461],{"id":1460},"beispiele","Beispiele",[564,1463,1464,1695,1696,2128,2129,2180,2181,2232,2233,2290,2291,2351,2352,2473,2474,1093],{},[573,1465,1467,1507],{"className":1466},[576],[573,1468,1470],{"className":1469},[580],[582,1471,1472],{"xmlns":584},[586,1473,1474,1504],{},[589,1475,1476,1493],{},[1477,1478,1479,1481,1490],"msubsup",{},[592,1480,594],{},[589,1482,1483,1485,1487],{},[599,1484,604],{},[592,1486,1252],{},[620,1488,1489],{},"1",[599,1491,1492],{"mathvariant":974},"∞",[1494,1495,1496,1502],"mfrac",{},[624,1497,1498,1500],{},[599,1499,611],{},[599,1501,604],{},[599,1503,604],{},[632,1505,1506],{"encoding":634},"\\sum_{n=1}^{\\infty} \\frac{x^n}{n}",[573,1508,1510],{"className":1509,"ariaHidden":640},[639],[573,1511,1513,1517,1584,1587],{"className":1512},[644],[573,1514],{"className":1515,"style":1516},[648],"height:1.256em;vertical-align:-0.345em;",[573,1518,1520,1523],{"className":1519},[653],[573,1521,594],{"className":1522,"style":656},[653,654,655],[573,1524,1526],{"className":1525},[673],[573,1527,1529,1575],{"className":1528},[677,678],[573,1530,1532,1572],{"className":1531},[682],[573,1533,1536,1557],{"className":1534,"style":1535},[686],"height:0.8043em;",[573,1537,1539,1542],{"style":1538},"top:-2.4003em;margin-left:0em;margin-right:0.05em;",[573,1540],{"className":1541,"style":695},[694],[573,1543,1545],{"className":1544},[699,700,701,702],[573,1546,1548,1551,1554],{"className":1547},[665,702],[573,1549,604],{"className":1550},[665,669,702],[573,1552,1252],{"className":1553},[1076,702],[573,1555,1489],{"className":1556},[665,702],[573,1558,1560,1563],{"style":1559},"top:-3.2029em;margin-right:0.05em;",[573,1561],{"className":1562,"style":695},[694],[573,1564,1566],{"className":1565},[699,700,701,702],[573,1567,1569],{"className":1568},[665,702],[573,1570,1492],{"className":1571},[665,702],[573,1573,710],{"className":1574},[709],[573,1576,1578],{"className":1577},[682],[573,1579,1582],{"className":1580,"style":1581},[686],"height:0.2997em;",[573,1583],{},[573,1585],{"className":1586,"style":661},[660],[573,1588,1590,1594,1692],{"className":1589},[665],[573,1591],{"className":1592},[723,1593],"nulldelimiter",[573,1595,1597],{"className":1596},[1494],[573,1598,1600,1683],{"className":1599},[677,678],[573,1601,1603,1680],{"className":1602},[682],[573,1604,1607,1623,1634],{"className":1605,"style":1606},[686],"height:0.911em;",[573,1608,1610,1614],{"style":1609},"top:-2.655em;",[573,1611],{"className":1612,"style":1613},[694],"height:3em;",[573,1615,1617],{"className":1616},[699,700,701,702],[573,1618,1620],{"className":1619},[665,702],[573,1621,604],{"className":1622},[665,669,702],[573,1624,1626,1629],{"style":1625},"top:-3.23em;",[573,1627],{"className":1628,"style":1613},[694],[573,1630],{"className":1631,"style":1633},[1632],"frac-line","border-bottom-width:0.04em;",[573,1635,1637,1640],{"style":1636},"top:-3.394em;",[573,1638],{"className":1639,"style":1613},[694],[573,1641,1643],{"className":1642},[699,700,701,702],[573,1644,1646],{"className":1645},[665,702],[573,1647,1649,1652],{"className":1648},[665,702],[573,1650,611],{"className":1651},[665,669,702],[573,1653,1655],{"className":1654},[673],[573,1656,1658],{"className":1657},[677],[573,1659,1661],{"className":1660},[682],[573,1662,1665],{"className":1663,"style":1664},[686],"height:0.7385em;",[573,1666,1668,1672],{"style":1667},"top:-2.931em;margin-right:0.0714em;",[573,1669],{"className":1670,"style":1671},[694],"height:2.5em;",[573,1673,1677],{"className":1674},[699,1675,1676,702],"reset-size3","size1",[573,1678,604],{"className":1679},[665,669,702],[573,1681,710],{"className":1682},[709],[573,1684,1686],{"className":1685},[682],[573,1687,1690],{"className":1688,"style":1689},[686],"height:0.345em;",[573,1691],{},[573,1693],{"className":1694},[788,1593],": Es ist ",[573,1697,1699,1757],{"className":1698},[576],[573,1700,1702],{"className":1701},[580],[582,1703,1704],{"xmlns":584},[586,1705,1706,1754],{},[589,1707,1708,1735,1737,1749,1752],{},[589,1709,1710,1712,1733],{},[592,1711,975],{"fence":640},[1494,1713,1714,1720],{},[596,1715,1716,1718],{},[599,1717,601],{},[599,1719,604],{},[596,1721,1722,1724],{},[599,1723,601],{},[589,1725,1726,1728,1731],{},[599,1727,604],{},[592,1729,1730],{},"+",[620,1732,1489],{},[592,1734,975],{"fence":640},[592,1736,1252],{},[1494,1738,1739,1747],{},[589,1740,1741,1743,1745],{},[599,1742,604],{},[592,1744,1730],{},[620,1746,1489],{},[599,1748,604],{},[592,1750,1751],{},"→",[620,1753,1489],{},[632,1755,1756],{"encoding":634},"\\left|\\frac{a_n}{a_{n+1}}\\right| = \\frac{n + 1}{n} \\to 1",[573,1758,1760,2027,2118],{"className":1759,"ariaHidden":640},[639],[573,1761,1763,1767,2018,2021,2024],{"className":1762},[644],[573,1764],{"className":1765,"style":1766},[648],"height:1.8em;vertical-align:-0.65em;",[573,1768,1771,1822,1981],{"className":1769},[1770],"minner",[573,1772,1774],{"className":1773},[723],[573,1775,1779],{"className":1776},[1777,1778],"delimsizing","mult",[573,1780,1782,1813],{"className":1781},[677,678],[573,1783,1785,1810],{"className":1784},[682],[573,1786,1789],{"className":1787,"style":1788},[686],"height:1.15em;",[573,1790,1792,1796],{"style":1791},"top:-3.15em;",[573,1793],{"className":1794,"style":1795},[694],"height:3.8em;",[573,1797,1799],{"style":1798},"width:0.333em;height:1.8em;",[1800,1801,1806],"svg",{"xmlns":1802,"width":1803,"height":1804,"viewBox":1805},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.333em","1.8em","0 0 333 1800",[1807,1808],"path",{"d":1809},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 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also ",[573,2130,2132,2150],{"className":2131},[576],[573,2133,2135],{"className":2134},[580],[582,2136,2137],{"xmlns":584},[586,2138,2139,2147],{},[589,2140,2141,2143,2145],{},[599,2142,937],{},[592,2144,1252],{},[620,2146,1489],{},[632,2148,2149],{"encoding":634},"r = 1",[573,2151,2153,2171],{"className":2152,"ariaHidden":640},[639],[573,2154,2156,2159,2162,2165,2168],{"className":2155},[644],[573,2157],{"className":2158,"style":843},[648],[573,2160,937],{"className":2161,"style":952},[665,669],[573,2163],{"className":2164,"style":1072},[660],[573,2166,1252],{"className":2167},[1076],[573,2169],{"className":2170,"style":1072},[660],[573,2172,2174,2177],{"className":2173},[644],[573,2175],{"className":2176,"style":2124},[648],[573,2178,1489],{"className":2179},[665],". 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Das Konvergenzintervall ist ",[573,2292,2294,2320],{"className":2293},[576],[573,2295,2297],{"className":2296},[580],[582,2298,2299],{"xmlns":584},[586,2300,2301,2317],{},[589,2302,2303,2306,2308,2310,2313,2315],{},[592,2304,2305],{"stretchy":607},"[",[592,2307,614],{},[620,2309,1489],{},[592,2311,2312],{"separator":640},",",[620,2314,1489],{},[592,2316,628],{"stretchy":607},[632,2318,2319],{"encoding":634},"[-1, 1)",[573,2321,2323],{"className":2322,"ariaHidden":640},[639],[573,2324,2326,2329,2332,2335,2338,2342,2345,2348],{"className":2325},[644],[573,2327],{"className":2328,"style":649},[648],[573,2330,2305],{"className":2331},[723],[573,2333,614],{"className":2334},[665],[573,2336,1489],{"className":2337},[665],[573,2339,2312],{"className":2340},[2341],"mpunct",[573,2343],{"className":2344,"style":661},[660],[573,2346,1489],{"className":2347},[665],[573,2349,628],{"className":2350},[788],". Für ",[573,2353,2355,2378],{"className":2354},[576],[573,2356,2358],{"className":2357},[580],[582,2359,2360],{"xmlns":584},[586,2361,2362,2375],{},[589,2363,2364,2366,2368],{},[599,2365,611],{},[592,2367,1252],{},[1494,2369,2370,2372],{},[620,2371,1489],{},[620,2373,2374],{},"2",[632,2376,2377],{"encoding":634},"x = \\frac{1}{2}",[573,2379,2381,2399],{"className":2380,"ariaHidden":640},[639],[573,2382,2384,2387,2390,2393,2396],{"className":2383},[644],[573,2385],{"className":2386,"style":843},[648],[573,2388,611],{"className":2389},[665,669],[573,2391],{"className":2392,"style":1072},[660],[573,2394,1252],{"className":2395},[1076],[573,2397],{"className":2398,"style":1072},[660],[573,2400,2402,2405],{"className":2401},[644],[573,2403],{"className":2404,"style":2033},[648],[573,2406,2408,2411,2470],{"className":2407},[665],[573,2409],{"className":2410},[723,1593],[573,2412,2414],{"className":2413},[1494],[573,2415,2417,2462],{"className":2416},[677,678],[573,2418,2420,2459],{"className":2419},[682],[573,2421,2423,2437,2445],{"className":2422,"style":2052},[686],[573,2424,2425,2428],{"style":1609},[573,2426],{"className":2427,"style":1613},[694],[573,2429,2431],{"className":2430},[699,700,701,702],[573,2432,2434],{"className":2433},[665,702],[573,2435,2374],{"className":2436},[665,702],[573,2438,2439,2442],{"style":1625},[573,2440],{"className":2441,"style":1613},[694],[573,2443],{"className":2444,"style":1633},[1632],[573,2446,2447,2450],{"style":1636},[573,2448],{"className":2449,"style":1613},[694],[573,2451,2453],{"className":2452},[699,700,701,702],[573,2454,2456],{"className":2455},[665,702],[573,2457,1489],{"className":2458},[665,702],[573,2460,710],{"className":2461},[709],[573,2463,2465],{"className":2464},[682],[573,2466,2468],{"className":2467,"style":1689},[686],[573,2469],{},[573,2471],{"className":2472},[788,1593]," 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x^n",[573,2565,2567],{"className":2566,"ariaHidden":640},[639],[573,2568,2570,2574,2637,2640,2737],{"className":2569},[644],[573,2571],{"className":2572,"style":2573},[648],"height:1.1493em;vertical-align:-0.345em;",[573,2575,2577,2580],{"className":2576},[653],[573,2578,594],{"className":2579,"style":656},[653,654,655],[573,2581,2583],{"className":2582},[673],[573,2584,2586,2629],{"className":2585},[677,678],[573,2587,2589,2626],{"className":2588},[682],[573,2590,2592,2612],{"className":2591,"style":1535},[686],[573,2593,2594,2597],{"style":1538},[573,2595],{"className":2596,"style":695},[694],[573,2598,2600],{"className":2599},[699,700,701,702],[573,2601,2603,2606,2609],{"className":2602},[665,702],[573,2604,604],{"className":2605},[665,669,702],[573,2607,1252],{"className":2608},[1076,702],[573,2610,1489],{"className":2611},[665,702],[573,2613,2614,2617],{"style":1559},[573,2615],{"className":2616,"style":695},[694],[573,2618,2620],{"className":2619},[699,700,701,702],[573,2621,2623],{"className":2622},[665,702],[573,2624,1492],{"className":2625},[665,702],[573,2627,710],{"className":2628},[709],[573,2630,2632],{"className":2631},[682],[573,2633,2635],{"className":2634,"style":1581},[686],[573,2636],{},[573,2638],{"className":2639,"style":661},[660],[573,2641,2643,2646,2734],{"className":2642},[665],[573,2644],{"className":2645},[723,1593],[573,2647,2649],{"className":2648},[1494],[573,2650,2652,2726],{"className":2651},[677,678],[573,2653,2655,2723],{"className":2654},[682],[573,2656,2659,2701,2709],{"className":2657,"style":2658},[686],"height:0.6954em;",[573,2660,2661,2664],{"style":1609},[573,2662],{"className":2663,"style":1613},[694],[573,2665,2667],{"className":2666},[699,700,701,702],[573,2668,2670],{"className":2669},[665,702],[573,2671,2673,2676],{"className":2672},[665,702],[573,2674,2552],{"className":2675},[665,702],[573,2677,2679],{"className":2678},[673],[573,2680,2682],{"className":2681},[677],[573,2683,2685],{"className":2684},[682],[573,2686,2689],{"className":2687,"style":2688},[686],"height:0.5935em;",[573,2690,2692,2695],{"style":2691},"top:-2.786em;margin-right:0.0714em;",[573,2693],{"className":2694,"style":1671},[694],[573,2696,2698],{"className":2697},[699,1675,1676,702],[573,2699,604],{"className":2700},[665,669,702],[573,2702,2703,2706],{"style":1625},[573,2704],{"className":2705,"style":1613},[694],[573,2707],{"className":2708,"style":1633},[1632],[573,2710,2711,2714],{"style":1636},[573,2712],{"className":2713,"style":1613},[694],[573,2715,2717],{"className":2716},[699,700,701,702],[573,2718,2720],{"className":2719},[665,702],[573,2721,604],{"className":2722},[665,669,702],[573,2724,710],{"className":2725},[709],[573,2727,2729],{"className":2728},[682],[573,2730,2732],{"className":2731,"style":1689},[686],[573,2733],{},[573,2735],{"className":2736},[788,1593],[573,2738,2740,2743],{"className":2739},[665],[573,2741,611],{"className":2742},[665,669],[573,2744,2746],{"className":2745},[673],[573,2747,2749],{"className":2748},[677],[573,2750,2752],{"className":2751},[682],[573,2753,2755],{"className":2754,"style":804},[686],[573,2756,2757,2760],{"style":807},[573,2758],{"className":2759,"style":695},[694],[573,2761,2763],{"className":2762},[699,700,701,702],[573,2764,604],{"className":2765},[665,669,702],": 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\\frac{a_n}{a_{n+1}} \\right| = \\frac{n \u002F 3^n}{(n + 1) \u002F 3^{n+1}} = \\frac{3n}{n + 1} \\to 3",[573,2879,2881,3132,3305,3400],{"className":2880,"ariaHidden":640},[639],[573,2882,2884,2888,3123,3126,3129],{"className":2883},[644],[573,2885],{"className":2886,"style":2887},[648],"height:2.4em;vertical-align:-0.95em;",[573,2889,2891,2936,3086],{"className":2890},[1770],[573,2892,2894],{"className":2893},[723],[573,2895,2897],{"className":2896},[1777,1778],[573,2898,2900,2927],{"className":2899},[677,678],[573,2901,2903,2924],{"className":2902},[682],[573,2904,2907],{"className":2905,"style":2906},[686],"height:1.45em;",[573,2908,2910,2914],{"style":2909},"top:-3.45em;",[573,2911],{"className":2912,"style":2913},[694],"height:4.4em;",[573,2915,2917],{"style":2916},"width:0.333em;height:2.4em;",[1800,2918,2921],{"xmlns":1802,"width":1803,"height":2919,"viewBox":2920},"2.4em","0 0 333 2400",[1807,2922],{"d":2923},"M145 15 v585 v1200 v585 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",[573,3413,3415,3433],{"className":3414},[576],[573,3416,3418],{"className":3417},[580],[582,3419,3420],{"xmlns":584},[586,3421,3422,3430],{},[589,3423,3424,3426,3428],{},[599,3425,937],{},[592,3427,1252],{},[620,3429,2552],{},[632,3431,3432],{"encoding":634},"r = 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Der Quotient ist ",[573,3777,3779,3801],{"className":3778},[576],[573,3780,3782],{"className":3781},[580],[582,3783,3784],{"xmlns":584},[586,3785,3786,3798],{},[589,3787,3788,3790,3792,3794,3796],{},[599,3789,604],{},[592,3791,1730],{},[620,3793,1489],{},[592,3795,1751],{},[599,3797,1492],{"mathvariant":974},[632,3799,3800],{"encoding":634},"n + 1 \\to \\infty",[573,3802,3804,3823,3841],{"className":3803,"ariaHidden":640},[639],[573,3805,3807,3811,3814,3817,3820],{"className":3806},[644],[573,3808],{"className":3809,"style":3810},[648],"height:0.6667em;vertical-align:-0.0833em;",[573,3812,604],{"className":3813},[665,669],[573,3815],{"className":3816,"style":730},[660],[573,3818,1730],{"className":3819},[734],[573,3821],{"className":3822,"style":730},[660],[573,3824,3826,3829,3832,3835,3838],{"className":3825},[644],[573,3827],{"className":3828,"style":2124},[648],[573,3830,1489],{"className":3831},[665],[573,3833],{"className":3834,"style":1072},[660],[573,3836,1751],{"className":3837},[1076],[573,3839],{"className":3840,"style":1072},[660],[573,3842,3844,3847],{"className":3843},[644],[573,3845],{"className":3846,"style":843},[648],[573,3848,1492],{"className":3849},[665],[573,3851,3853,3871],{"className":3852},[576],[573,3854,3856],{"className":3855},[580],[582,3857,3858],{"xmlns":584},[586,3859,3860,3868],{},[589,3861,3862,3864,3866],{},[599,3863,937],{},[592,3865,1252],{},[599,3867,1492],{"mathvariant":974},[632,3869,3870],{"encoding":634},"r = \\infty",[573,3872,3874,3892],{"className":3873,"ariaHidden":640},[639],[573,3875,3877,3880,3883,3886,3889],{"className":3876},[644],[573,3878],{"className":3879,"style":843},[648],[573,3881,937],{"className":3882,"style":952},[665,669],[573,3884],{"className":3885,"style":1072},[660],[573,3887,1252],{"className":3888},[1076],[573,3890],{"className":3891,"style":1072},[660],[573,3893,3895,3898],{"className":3894},[644],[573,3896],{"className":3897,"style":843},[648],[573,3899,1492],{"className":3900},[665],". Die Exponentialreihe konvergiert für alle ",[573,3903,3905,3918],{"className":3904},[576],[573,3906,3908],{"className":3907},[580],[582,3909,3910],{"xmlns":584},[586,3911,3912,3916],{},[589,3913,3914],{},[599,3915,611],{},[632,3917,611],{"encoding":634},[573,3919,3921],{"className":3920,"ariaHidden":640},[639],[573,3922,3924,3927],{"className":3923},[644],[573,3925],{"className":3926,"style":843},[648],[573,3928,611],{"className":3929},[665,669],[559,3931,3933],{"id":3932},"typische-fehler","Typische Fehler",[955,3935,3936,4463,4466],{},[958,3937,3938,3939,4138,4139,4335,4336,4434,4435,1093],{},"Den Quotienten verkehrt herum bilden. Beim Quotientenkriterium steht ",[573,3940,3942,3974],{"className":3941},[576],[573,3943,3945],{"className":3944},[580],[582,3946,3947],{"xmlns":584},[586,3948,3949,3971],{},[589,3950,3951],{},[1494,3952,3953,3965],{},[596,3954,3955,3957],{},[599,3956,601],{},[589,3958,3959,3961,3963],{},[599,3960,604],{},[592,3962,1730],{},[620,3964,1489],{},[596,3966,3967,3969],{},[599,3968,601],{},[599,3970,604],{},[632,3972,3973],{"encoding":634},"\\frac{a_{n+1}}{a_n}",[573,3975,3977],{"className":3976,"ariaHidden":640},[639],[573,3978,3980,3984],{"className":3979},[644],[573,3981],{"className":3982,"style":3983},[648],"height:1.1983em;vertical-align:-0.4451em;",[573,3985,3987,3990,4135],{"className":3986},[665],[573,3988],{"className":3989},[723,1593],[573,3991,3993],{"className":3992},[1494],[573,3994,3996,4126],{"className":3995},[677,678],[573,3997,3999,4123],{"className":3998},[682],[573,4000,4003,4054,4062],{"className":4001,"style":4002},[686],"height:0.7532em;",[573,4004,4005,4008],{"style":1609},[573,4006],{"className":4007,"style":1613},[694],[573,4009,4011],{"className":4010},[699,700,701,702],[573,4012,4014],{"className":4013},[665,702],[573,4015,4017,4020],{"className":4016},[665,702],[573,4018,601],{"className":4019},[665,669,702],[573,4021,4023],{"className":4022},[673],[573,4024,4026,4046],{"className":4025},[677,678],[573,4027,4029,4043],{"className":4028},[682],[573,4030,4032],{"className":4031,"style":1942},[686],[573,4033,4034,4037],{"style":1873},[573,4035],{"className":4036,"style":1671},[694],[573,4038,4040],{"className":4039},[699,1675,1676,702],[573,4041,604],{"className":4042},[665,669,702],[573,4044,710],{"className":4045},[709],[573,4047,4049],{"className":4048},[682],[573,4050,4052],{"className":4051,"style":1963},[686],[573,4053],{},[573,4055,4056,4059],{"style":1625},[573,4057],{"className":4058,"style":1613},[694],[573,4060],{"className":4061,"style":1633},[1632],[573,4063,4065,4068],{"style":4064},"top:-3.4518em;",[573,4066],{"className":4067,"style":1613},[694],[573,4069,4071],{"className":4070},[699,700,701,702],[573,4072,4074],{"className":4073},[665,702],[573,4075,4077,4080],{"className":4076},[665,702],[573,4078,601],{"className":4079},[665,669,702],[573,4081,4083],{"className":4082},[673],[573,4084,4086,4115],{"className":4085},[677,678],[573,4087,4089,4112],{"className":4088},[682],[573,4090,4092],{"className":4091,"style":1870},[686],[573,4093,4094,4097],{"style":1873},[573,4095],{"className":4096,"style":1671},[694],[573,4098,4100],{"className":4099},[699,1675,1676,702],[573,4101,4103,4106,4109],{"className":4102},[665,702],[573,4104,604],{"className":4105},[665,669,702],[573,4107,1730],{"className":4108},[734,702],[573,4110,1489],{"className":4111},[665,702],[573,4113,710],{"className":4114},[709],[573,4116,4118],{"className":4117},[682],[573,4119,4121],{"className":4120,"style":1901},[686],[573,4122],{},[573,4124,710],{"className":4125},[709],[573,4127,4129],{"className":4128},[682],[573,4130,4133],{"className":4131,"style":4132},[686],"height:0.4451em;",[573,4134],{},[573,4136],{"className":4137},[788,1593],", für den Radius ",[573,4140,4142,4174],{"className":4141},[576],[573,4143,4145],{"className":4144},[580],[582,4146,4147],{"xmlns":584},[586,4148,4149,4171],{},[589,4150,4151],{},[1494,4152,4153,4159],{},[596,4154,4155,4157],{},[599,4156,601],{},[599,4158,604],{},[596,4160,4161,4163],{},[599,4162,601],{},[589,4164,4165,4167,4169],{},[599,4166,604],{},[592,4168,1730],{},[620,4170,1489],{},[632,4172,4173],{"encoding":634},"\\frac{a_n}{a_{n+1}}",[573,4175,4177],{"className":4176,"ariaHidden":640},[639],[573,4178,4180,4184],{"className":4179},[644],[573,4181],{"className":4182,"style":4183},[648],"height:1.1983em;vertical-align:-0.4868em;",[573,4185,4187,4190,4332],{"className":4186},[665],[573,4188],{"className":4189},[723,1593],[573,4191,4193],{"className":4192},[1494],[573,4194,4196,4324],{"className":4195},[677,678],[573,4197,4199,4321],{"className":4198},[682],[573,4200,4202,4262,4270],{"className":4201,"style":1840},[686],[573,4203,4204,4207],{"style":1609},[573,4205],{"className":4206,"style":1613},[694],[573,4208,4210],{"className":4209},[699,700,701,702],[573,4211,4213],{"className":4212},[665,702],[573,4214,4216,4219],{"className":4215},[665,702],[573,4217,601],{"className":4218},[665,669,702],[573,4220,4222],{"className":4221},[673],[573,4223,4225,4254],{"className":4224},[677,678],[573,4226,4228,4251],{"className":4227},[682],[573,4229,4231],{"className":4230,"style":1870},[686],[573,4232,4233,4236],{"style":1873},[573,4234],{"className":4235,"style":1671},[694],[573,4237,4239],{"className":4238},[699,1675,1676,702],[573,4240,4242,4245,4248],{"className":4241},[665,702],[573,4243,604],{"className":4244},[665,669,702],[573,4246,1730],{"className":4247},[734,702],[573,4249,1489],{"className":4250},[665,702],[573,4252,710],{"className":4253},[709],[573,4255,4257],{"className":4256},[682],[573,4258,4260],{"className":4259,"style":1901},[686],[573,4261],{},[573,4263,4264,4267],{"style":1625},[573,4265],{"className":4266,"style":1613},[694],[573,4268],{"className":4269,"style":1633},[1632],[573,4271,4272,4275],{"style":1914},[573,4273],{"className":4274,"style":1613},[694],[573,4276,4278],{"className":4277},[699,700,701,702],[573,4279,4281],{"className":4280},[665,702],[573,4282,4284,4287],{"className":4283},[665,702],[573,4285,601],{"className":4286},[665,669,702],[573,4288,4290],{"className":4289},[673],[573,4291,4293,4313],{"className":4292},[677,678],[573,4294,4296,4310],{"className":4295},[682],[573,4297,4299],{"className":4298,"style":1942},[686],[573,4300,4301,4304],{"style":1873},[573,4302],{"className":4303,"style":1671},[694],[573,4305,4307],{"className":4306},[699,1675,1676,702],[573,4308,604],{"className":4309},[665,669,702],[573,4311,710],{"className":4312},[709],[573,4314,4316],{"className":4315},[682],[573,4317,4319],{"className":4318,"style":1963},[686],[573,4320],{},[573,4322,710],{"className":4323},[709],[573,4325,4327],{"className":4326},[682],[573,4328,4330],{"className":4329,"style":1975},[686],[573,4331],{},[573,4333],{"className":4334},[788,1593],". Im zweiten Beispiel ergäbe der falsche Quotient ",[573,4337,4339,4357],{"className":4338},[576],[573,4340,4342],{"className":4341},[580],[582,4343,4344],{"xmlns":584},[586,4345,4346,4354],{},[589,4347,4348],{},[1494,4349,4350,4352],{},[620,4351,1489],{},[620,4353,2552],{},[632,4355,4356],{"encoding":634},"\\frac{1}{3}",[573,4358,4360],{"className":4359,"ariaHidden":640},[639],[573,4361,4363,4366],{"className":4362},[644],[573,4364],{"className":4365,"style":2033},[648],[573,4367,4369,4372,4431],{"className":4368},[665],[573,4370],{"className":4371},[723,1593],[573,4373,4375],{"className":4374},[1494],[573,4376,4378,4423],{"className":4377},[677,678],[573,4379,4381,4420],{"className":4380},[682],[573,4382,4384,4398,4406],{"className":4383,"style":2052},[686],[573,4385,4386,4389],{"style":1609},[573,4387],{"className":4388,"style":1613},[694],[573,4390,4392],{"className":4391},[699,700,701,702],[573,4393,4395],{"className":4394},[665,702],[573,4396,2552],{"className":4397},[665,702],[573,4399,4400,4403],{"style":1625},[573,4401],{"className":4402,"style":1613},[694],[573,4404],{"className":4405,"style":1633},[1632],[573,4407,4408,4411],{"style":1636},[573,4409],{"className":4410,"style":1613},[694],[573,4412,4414],{"className":4413},[699,700,701,702],[573,4415,4417],{"className":4416},[665,702],[573,4418,1489],{"className":4419},[665,702],[573,4421,710],{"className":4422},[709],[573,4424,4426],{"className":4425},[682],[573,4427,4429],{"className":4428,"style":1689},[686],[573,4430],{},[573,4432],{"className":4433},[788,1593]," statt ",[573,4436,4438,4451],{"className":4437},[576],[573,4439,4441],{"className":4440},[580],[582,4442,4443],{"xmlns":584},[586,4444,4445,4449],{},[589,4446,4447],{},[620,4448,2552],{},[632,4450,2552],{"encoding":634},[573,4452,4454],{"className":4453,"ariaHidden":640},[639],[573,4455,4457,4460],{"className":4456},[644],[573,4458],{"className":4459,"style":2124},[648],[573,4461,2552],{"className":4462},[665],[958,4464,4465],{},"Die Randpunkte vergessen oder mit innen gleichsetzen. Wie das erste Beispiel zeigt, kann die Reihe an einem Rand konvergieren und am anderen divergieren.",[958,4467,4468,4469,4497,4498,4567,4568,4729,4730,1093],{},"Das Intervall um ",[573,4470,4472,4485],{"className":4471},[576],[573,4473,4475],{"className":4474},[580],[582,4476,4477],{"xmlns":584},[586,4478,4479,4483],{},[589,4480,4481],{},[620,4482,622],{},[632,4484,622],{"encoding":634},[573,4486,4488],{"className":4487,"ariaHidden":640},[639],[573,4489,4491,4494],{"className":4490},[644],[573,4492],{"className":4493,"style":2124},[648],[573,4495,622],{"className":4496},[665]," statt um ",[573,4499,4501,4518],{"className":4500},[576],[573,4502,4504],{"className":4503},[580],[582,4505,4506],{"xmlns":584},[586,4507,4508,4516],{},[589,4509,4510],{},[596,4511,4512,4514],{},[599,4513,611],{},[620,4515,622],{},[632,4517,868],{"encoding":634},[573,4519,4521],{"className":4520,"ariaHidden":640},[639],[573,4522,4524,4527],{"className":4523},[644],[573,4525],{"className":4526,"style":878},[648],[573,4528,4530,4533],{"className":4529},[665],[573,4531,611],{"className":4532},[665,669],[573,4534,4536],{"className":4535},[673],[573,4537,4539,4559],{"className":4538},[677,678],[573,4540,4542,4556],{"className":4541},[682],[573,4543,4545],{"className":4544,"style":762},[686],[573,4546,4547,4550],{"style":690},[573,4548],{"className":4549,"style":695},[694],[573,4551,4553],{"className":4552},[699,700,701,702],[573,4554,622],{"className":4555},[665,702],[573,4557,710],{"className":4558},[709],[573,4560,4562],{"className":4561},[682],[573,4563,4565],{"className":4564,"style":717},[686],[573,4566],{}," legen. Bei ",[573,4569,4571,4605],{"className":4570},[576],[573,4572,4574],{"className":4573},[580],[582,4575,4576],{"xmlns":584},[586,4577,4578,4602],{},[589,4579,4580,4582],{},[592,4581,594],{},[1494,4583,4584,4600],{},[589,4585,4586,4588,4590,4592,4594],{},[592,4587,608],{"stretchy":607},[599,4589,611],{},[592,4591,614],{},[620,4593,2374],{},[624,4595,4596,4598],{},[592,4597,628],{"stretchy":607},[599,4599,604],{},[599,4601,604],{},[632,4603,4604],{"encoding":634},"\\sum \\frac{(x - 2)^n}{n}",[573,4606,4608],{"className":4607,"ariaHidden":640},[639],[573,4609,4611,4615,4618,4621],{"className":4610},[644],[573,4612],{"className":4613,"style":4614},[648],"height:1.355em;vertical-align:-0.345em;",[573,4616,594],{"className":4617,"style":656},[653,654,655],[573,4619],{"className":4620,"style":661},[660],[573,4622,4624,4627,4726],{"className":4623},[665],[573,4625],{"className":4626},[723,1593],[573,4628,4630],{"className":4629},[1494],[573,4631,4633,4718],{"className":4632},[677,678],[573,4634,4636,4715],{"className":4635},[682],[573,4637,4640,4654,4662],{"className":4638,"style":4639},[686],"height:1.01em;",[573,4641,4642,4645],{"style":1609},[573,4643],{"className":4644,"style":1613},[694],[573,4646,4648],{"className":4647},[699,700,701,702],[573,4649,4651],{"className":4650},[665,702],[573,4652,604],{"className":4653},[665,669,702],[573,4655,4656,4659],{"style":1625},[573,4657],{"className":4658,"style":1613},[694],[573,4660],{"className":4661,"style":1633},[1632],[573,4663,4665,4668],{"style":4664},"top:-3.485em;",[573,4666],{"className":4667,"style":1613},[694],[573,4669,4671],{"className":4670},[699,700,701,702],[573,4672,4674,4677,4680,4683,4686],{"className":4673},[665,702],[573,4675,608],{"className":4676},[723,702],[573,4678,611],{"className":4679},[665,669,702],[573,4681,614],{"className":4682},[734,702],[573,4684,2374],{"className":4685},[665,702],[573,4687,4689,4692],{"className":4688},[788,702],[573,4690,628],{"className":4691},[788,702],[573,4693,4695],{"className":4694},[673],[573,4696,4698],{"className":4697},[677],[573,4699,4701],{"className":4700},[682],[573,4702,4704],{"className":4703,"style":1664},[686],[573,4705,4706,4709],{"style":1667},[573,4707],{"className":4708,"style":1671},[694],[573,4710,4712],{"className":4711},[699,1675,1676,702],[573,4713,604],{"className":4714},[665,669,702],[573,4716,710],{"className":4717},[709],[573,4719,4721],{"className":4720},[682],[573,4722,4724],{"className":4723,"style":1689},[686],[573,4725],{},[573,4727],{"className":4728},[788,1593]," ist das Konvergenzintervall ",[573,4731,4733,4755],{"className":4732},[576],[573,4734,4736],{"className":4735},[580],[582,4737,4738],{"xmlns":584},[586,4739,4740,4752],{},[589,4741,4742,4744,4746,4748,4750],{},[592,4743,2305],{"stretchy":607},[620,4745,1489],{},[592,4747,2312],{"separator":640},[620,4749,2552],{},[592,4751,628],{"stretchy":607},[632,4753,4754],{"encoding":634},"[1, 3)",[573,4756,4758],{"className":4757,"ariaHidden":640},[639],[573,4759,4761,4764,4767,4770,4773,4776,4779],{"className":4760},[644],[573,4762],{"className":4763,"style":649},[648],[573,4765,2305],{"className":4766},[723],[573,4768,1489],{"className":4769},[665],[573,4771,2312],{"className":4772},[2341],[573,4774],{"className":4775,"style":661},[660],[573,4777,2552],{"className":4778},[665],[573,4780,628],{"className":4781},[788],{"title":4783,"searchDepth":4784,"depth":4784,"links":4785},"",2,[4786,4787,4788],{"id":561,"depth":4784,"text":562},{"id":1460,"depth":4784,"text":1461},{"id":3932,"depth":4784,"text":3933},[4790,4793,4796,4799,4802,4807],{"lhs":4791,"rhs":4792},"(1\u002Fn)\u002F(1\u002F(n + 1))","(n + 1)\u002Fn",{"lhs":4794,"rhs":4795},"round(sum(map(1:60, f(n) = 0.5^n\u002Fn)), 9)","round(log(2), 9)",{"lhs":4797,"rhs":4798},"(n\u002F3^n)\u002F((n + 1)\u002F3^(n + 1))","3*n\u002F(n + 1)",{"lhs":4800,"rhs":4801},"round(3*1000\u002F1001, 3)","2.997",{"lhs":4803,"rhs":4804,"where":4805},"(1\u002Fn!)\u002F(1\u002F(n + 1)!)","n + 1",{"n":4806},6,{"lhs":4808,"rhs":4809},"((n + 1)\u002F3^(n + 1))\u002F(n\u002F3^n)","(n + 1)\u002F(3*n)","\\text{Potenzreihe } \\sum_{n=0}^{\\infty} a_n (x - x_0)^n;\\ \\text{zweite Form nur, wenn } a_n \\neq 0 \\text{ und der Grenzwert existiert};\\ \\tfrac{1}{0} := \\infty,\\ \\tfrac{1}{\\infty} := 0","\u003Cspan class=\"katex\">Potenzreihe\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsubsup>\u003Cmo>∑\u003C\u002Fmo>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmsubsup>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmi>x\u003C\u002Fmi>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsup>\u003Cmo separator=\"true\">;\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\sum_{n=0}^{\\infty} a_n (x - x_0)^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:1.104em;vertical-align:-0.2997em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">\u003Cspan class=\"mop op-symbol small-op\" style=\"position:relative;top:0em;\">∑\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8043em;\">\u003Cspan style=\"top:-2.4003em;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 mtight\">\u003Cspan class=\"mord mathnormal mtight\">n\u003C\u002Fspan>\u003Cspan class=\"mrel mtight\">=\u003C\u002Fspan>\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2029em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">∞\u003C\u002Fspan>\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:0.2997em;\">\u003Cspan>\u003C\u002Fspan>\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=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\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 mtight\">0\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\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6644em;\">\u003Cspan style=\"top:-3.063em;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>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">;\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">zweite Form nur, wenn\u003C\u002Fspan> \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>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">a_n \\neq 0\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:0.8889em;vertical-align:-0.1944em;\">\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\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">und der Grenzwert\u003C\u002Fspan> \u003Cspan style=\"white-space: nowrap\">\u003Cspan class=\"katex\">existiert\u003C\u002Fspan>\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmo separator=\"true\">;\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">;\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=\"mpunct\">;\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmfrac>\u003C\u002Fmstyle>\u003Cmo>:\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tfrac{1}{0} := \\infty,\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:1.1901em;vertical-align:-0.345em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\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:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">∞\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmstyle scriptlevel=\"0\" displaystyle=\"false\">\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmfrac>\u003C\u002Fmstyle>\u003Cmo>:\u003C\u002Fmo>\u003Cmo>=\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\tfrac{1}{\\infty} := 0\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:1.1901em;vertical-align:-0.345em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8451em;\">\u003Cspan style=\"top:-2.655em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">∞\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.23em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"frac-line\" style=\"border-bottom-width:0.04em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.394em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\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:0.345em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose nulldelimiter\">\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:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","radius-of-convergence","r = \\frac{1}{\\limsup\\limits_{n \\to \\infty} \\sqrt[n]{|a_n|}} \\quad\\text{bzw.}\\quad r = \\lim_{n \\to \\infty} \\left| \\frac{a_n}{a_{n+1}} \\right|","\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>\u003Cmi>r\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\u003Cmn>1\u003C\u002Fmn>\u003Cmrow>\u003Cmunder>\u003Cmrow>\u003Cmi mathvariant=\"normal\">lim sup\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>→\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∞\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmunder>\u003Cmroot>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmroot>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003Cmspace width=\"1em\"\u002F>\u003Cmtext>bzw.\u003C\u002Fmtext>\u003Cmspace width=\"1em\"\u002F>\u003Cmi>r\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\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>\u003Cmrow>\u003Cmo fence=\"true\">∣\u003C\u002Fmo>\u003Cmfrac>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmi>n\u003C\u002Fmi>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmrow>\u003Cmi>n\u003C\u002Fmi>\u003Cmo>+\u003C\u002Fmo>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmrow>\u003C\u002Fmsub>\u003C\u002Fmfrac>\u003Cmo fence=\"true\">∣\u003C\u002Fmo>\u003C\u002Fmrow>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r = \\frac{1}{\\limsup\\limits_{n \\to \\infty} \\sqrt[n]{|a_n|}} \\quad\\text{bzw.}\\quad r = \\lim_{n \\to \\infty} \\left| \\frac{a_n}{a_{n+1}} \\right|\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan 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