40:
1869:
2178:
872:
672:
367:
1477:
527:
1197:
1027:
1297:
867:{\displaystyle {\begin{aligned}\sum _{k=0}^{\infty }(-1)^{k}k!&=\int _{0}^{\infty }{\frac {e^{-x}}{1+x}}\,dx\\&=eE_{1}(1)\approx 0.596\,347\,362\,323\,194\,074\,341\,078\,499\,369\ldots \end{aligned}}}
677:
163:
218:
2060:
644:
603:
1318:). Euler argues (more or less) that since the formal series and the integral both describe the same solution to the differential equations, they should equal each other at
1350:
566:
2050:
1082:
1703:
1342:
664:
2143:
1984:
378:
1090:
1994:
914:
1215:
1989:
2158:
1749:
1696:
2138:
2040:
2030:
1519:
1514:
83:
61:
2148:
2153:
2055:
1689:
2181:
2163:
105:
2035:
2025:
2015:
1509:
1504:
1493:
2045:
2202:
362:{\displaystyle \sum _{k=0}^{\infty }(-1)^{k}k!=\sum _{k=0}^{\infty }(-1)^{k}\int _{0}^{\infty }x^{k}e^{-x}\,dx.}
54:
48:
2130:
1952:
1792:
1739:
1590:
65:
208:
that are alternately added or subtracted. One way to assign a value to this divergent series is by using
184:
with alternating signs. Despite being divergent, it can be assigned a value of approximately 0.596347 by
1999:
1744:
1488:
606:
28:
612:
571:
2110:
1947:
1716:
1635:
1560:
1307:
1303:
889:
1595:
2090:
1957:
372:
If summation and integration are interchanged (ignoring that neither side converges), one obtains:
1472:{\displaystyle \sum _{n=1}^{\infty }(-1)^{n+1}(n-1)!=e\int _{1}^{\infty }{\frac {e^{-u}}{u}}\,du.}
1931:
1916:
1888:
1868:
1807:
1651:
1608:
1550:
201:
2020:
1498:
2120:
1921:
1893:
1847:
1837:
1817:
1802:
897:
535:
2105:
1926:
1852:
1842:
1822:
1724:
1671:
1643:
1600:
169:
17:
1883:
1812:
209:
185:
1321:
1639:
1564:
1542:
2115:
2100:
2095:
1774:
1759:
1538:
649:
197:
181:
173:
2196:
2080:
1754:
1675:
1655:
1620:
1647:
2085:
1827:
1769:
522:{\displaystyle \sum _{k=0}^{\infty }(-1)^{k}k!=\int _{0}^{\infty }\lefte^{-x}\,dx.}
1662:
Leah, P. J.; Barbeau, E. J. (May 1976), "Euler's 1760 paper on divergent series",
1832:
1779:
1764:
893:
205:
177:
1192:{\displaystyle x(t)=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {(n-1)!}{t^{n}}}}
1712:
1681:
1612:
1209:
On the other hand, the system of differential equations has a solution
1022:{\displaystyle {\dot {x}}(t)=x(t)-y(t),\qquad {\dot {y}}(t)=-y(t)^{2}}
1292:{\displaystyle x(t)=e^{t}\int _{t}^{\infty }{\frac {e^{-u}}{u}}\,du.}
1604:
1555:
1685:
204:
to assign a finite value to the series. The series is a sum of
33:
1581:
Kline, Morris (November 1983), "Euler and
Infinite Series",
27:
For a related alternating partial sum of factorials, see
1081:, and substituting it into the first equation gives a
908:
Consider the coupled system of differential equations
617:
576:
96:
Divergent series that can be summed by Borel summation
2061:
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
2051:
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
1547:
1353:
1324:
1218:
1093:
917:
675:
652:
615:
574:
538:
381:
221:
108:
532:
The summation in the square brackets converges when
2129:
2073:
2008:
1977:
1970:
1940:
1909:
1902:
1876:
1788:
1732:
1723:
1471:
1336:
1291:
1191:
1021:
866:
666:leads to a convergent integral for the summation:
658:
638:
597:
560:
521:
361:
157:
158:{\displaystyle \sum _{k=0}^{\infty }(-1)^{k}k!}
1306:, the formal power series is recovered as an
1032:where dots denote derivatives with respect to
1697:
1621:"Euler and mathematical methods in mechanics"
8:
2144:Hypergeometric function of a matrix argument
2000:1 + 1/2 + 1/3 + ... (Riemann zeta function)
1974:
1906:
1729:
1704:
1690:
1682:
2056:1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
1594:
1554:
1459:
1445:
1439:
1433:
1428:
1388:
1369:
1358:
1352:
1323:
1279:
1265:
1259:
1253:
1248:
1238:
1217:
1181:
1155:
1143:
1124:
1113:
1092:
1013:
974:
973:
919:
918:
916:
853:
849:
845:
841:
837:
833:
829:
825:
821:
800:
776:
754:
748:
742:
737:
714:
695:
684:
676:
674:
651:
616:
614:
575:
573:
547:
539:
537:
509:
500:
485:
466:
455:
440:
435:
416:
397:
386:
380:
349:
340:
330:
320:
315:
305:
286:
275:
256:
237:
226:
220:
143:
124:
113:
107:
84:Learn how and when to remove this message
1039:The solution with stable equilibrium at
47:This article includes a list of general
1530:
7:
904:Connection to differential equations
196:This series was first considered by
2021:1 − 1 + 1 − 1 + ⋯ (Grandi's series)
896:of the series, and is equal to the
1434:
1370:
1254:
1125:
743:
696:
467:
441:
398:
321:
287:
238:
125:
53:it lacks sufficient corresponding
25:
2139:Generalized hypergeometric series
1206:(1) is precisely Euler's series.
639:{\displaystyle {\tfrac {1}{1+x}}}
598:{\displaystyle {\tfrac {1}{1+x}}}
2177:
2176:
2149:Lauricella hypergeometric series
1867:
38:
2159:Riemann's differential equation
1648:10.1070/rm2007v062n04abeh004427
1545:[On divergent series].
972:
1412:
1400:
1385:
1375:
1228:
1222:
1170:
1158:
1140:
1130:
1103:
1097:
1010:
1003:
991:
985:
966:
960:
951:
945:
936:
930:
812:
806:
711:
701:
568:, and for those values equals
548:
540:
482:
472:
413:
403:
302:
292:
253:
243:
140:
130:
1:
2154:Modular hypergeometric series
1995:1/4 + 1/16 + 1/64 + 1/256 + ⋯
1676:10.1016/0315-0860(76)90030-6
1628:Russian Mathematical Surveys
892:. This is by definition the
212:, where one formally writes
18:1 − 1 + 2 − 6 + 24 − 120 + …
2164:Theta hypergeometric series
1543:"De seriebus divergentibus"
2219:
2046:Infinite arithmetic series
1990:1/2 + 1/4 + 1/8 + 1/16 + ⋯
1985:1/2 − 1/4 + 1/8 − 1/16 + ⋯
26:
2172:
1865:
192:Euler and Borel summation
1308:asymptotic approximation
561:{\displaystyle |x|<1}
1877:Properties of sequences
1310:to this expression for
68:more precise citations.
1740:Arithmetic progression
1619:Kozlov, V. V. (2007),
1473:
1374:
1338:
1293:
1193:
1129:
1083:formal series solution
1023:
868:
700:
660:
640:
599:
562:
523:
471:
402:
363:
291:
242:
172:, first considered by
159:
129:
2131:Hypergeometric series
1745:Geometric progression
1489:Alternating factorial
1474:
1354:
1339:
1294:
1194:
1109:
1024:
869:
680:
661:
646:to all positive real
641:
607:analytic continuation
600:
563:
524:
451:
382:
364:
271:
222:
160:
109:
29:Alternating factorial
2111:Trigonometric series
1903:Properties of series
1750:Harmonic progression
1664:Historia Mathematica
1583:Mathematics Magazine
1351:
1322:
1304:integrating by parts
1216:
1091:
915:
890:exponential integral
673:
650:
613:
572:
536:
379:
219:
106:
2091:Formal power series
1640:2007RuMaS..62..639K
1565:2018arXiv180802841E
1438:
1337:{\displaystyle t=1}
1258:
1055: → ∞ has
747:
445:
325:
202:summability methods
1889:Monotonic function
1808:Fibonacci sequence
1469:
1424:
1334:
1289:
1244:
1189:
1019:
864:
862:
733:
656:
636:
634:
595:
593:
558:
519:
431:
359:
311:
155:
2190:
2189:
2121:Generating series
2069:
2068:
2041:1 − 2 + 4 − 8 + ⋯
2036:1 + 2 + 4 + 8 + ⋯
2031:1 − 2 + 3 − 4 + ⋯
2026:1 + 2 + 3 + 4 + ⋯
2016:1 + 1 + 1 + 1 + ⋯
1966:
1965:
1894:Periodic sequence
1863:
1862:
1848:Triangular number
1838:Pentagonal number
1818:Heptagonal number
1803:Complete sequence
1725:Integer sequences
1520:1 − 2 + 4 − 8 + ⋯
1515:1 − 2 + 3 − 4 + ⋯
1510:1 + 2 + 4 + 8 + ⋯
1505:1 + 2 + 3 + 4 + ⋯
1499:1 − 1 + 1 − 1 + ⋯
1494:1 + 1 + 1 + 1 + ⋯
1457:
1277:
1187:
982:
927:
898:Gompertz constant
774:
659:{\displaystyle x}
633:
592:
99:In mathematics,
94:
93:
86:
16:(Redirected from
2210:
2203:Divergent series
2180:
2179:
2106:Dirichlet series
1975:
1907:
1871:
1843:Polygonal number
1823:Hexagonal number
1796:
1730:
1706:
1699:
1692:
1683:
1678:
1658:
1625:
1615:
1598:
1569:
1568:
1558:
1535:
1478:
1476:
1475:
1470:
1458:
1453:
1452:
1440:
1437:
1432:
1399:
1398:
1373:
1368:
1343:
1341:
1340:
1335:
1302:By successively
1298:
1296:
1295:
1290:
1278:
1273:
1272:
1260:
1257:
1252:
1243:
1242:
1198:
1196:
1195:
1190:
1188:
1186:
1185:
1176:
1156:
1154:
1153:
1128:
1123:
1080:
1078:
1077:
1072:
1069:
1050:
1028:
1026:
1025:
1020:
1018:
1017:
984:
983:
975:
929:
928:
920:
873:
871:
870:
865:
863:
805:
804:
786:
775:
773:
762:
761:
749:
746:
741:
719:
718:
699:
694:
665:
663:
662:
657:
645:
643:
642:
637:
635:
632:
618:
604:
602:
601:
596:
594:
591:
577:
567:
565:
564:
559:
551:
543:
528:
526:
525:
520:
508:
507:
495:
491:
490:
489:
470:
465:
444:
439:
421:
420:
401:
396:
368:
366:
365:
360:
348:
347:
335:
334:
324:
319:
310:
309:
290:
285:
261:
260:
241:
236:
176:, that sums the
170:divergent series
164:
162:
161:
156:
148:
147:
128:
123:
89:
82:
78:
75:
69:
64:this article by
55:inline citations
42:
41:
34:
21:
2218:
2217:
2213:
2212:
2211:
2209:
2208:
2207:
2193:
2192:
2191:
2186:
2168:
2125:
2074:Kinds of series
2065:
2004:
1971:Explicit series
1962:
1936:
1898:
1884:Cauchy sequence
1872:
1859:
1813:Figurate number
1790:
1784:
1775:Powers of three
1719:
1710:
1661:
1623:
1618:
1605:10.2307/2690371
1596:10.1.1.639.6923
1580:
1577:
1575:Further reading
1572:
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1528:
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533:
496:
481:
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377:
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336:
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301:
252:
217:
216:
210:Borel summation
194:
186:Borel summation
182:natural numbers
139:
104:
103:
97:
90:
79:
73:
70:
60:Please help to
59:
43:
39:
32:
23:
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2156:
2151:
2146:
2141:
2135:
2133:
2127:
2126:
2124:
2123:
2118:
2116:Fourier series
2113:
2108:
2103:
2101:Puiseux series
2098:
2096:Laurent series
2093:
2088:
2083:
2077:
2075:
2071:
2070:
2067:
2066:
2064:
2063:
2058:
2053:
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2023:
2018:
2012:
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1997:
1992:
1987:
1981:
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1972:
1968:
1967:
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1805:
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1734:
1727:
1721:
1720:
1711:
1709:
1708:
1701:
1694:
1686:
1680:
1679:
1670:(2): 141–160,
1659:
1634:(4): 639–661,
1616:
1589:(5): 307–313,
1576:
1573:
1571:
1570:
1549:(5): 205–237.
1529:
1527:
1524:
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1517:
1512:
1507:
1502:
1496:
1491:
1484:
1481:
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1468:
1465:
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1444:
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1427:
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1411:
1408:
1405:
1402:
1397:
1394:
1391:
1387:
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1367:
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1361:
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1330:
1327:
1300:
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1271:
1268:
1264:
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1247:
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1237:
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1227:
1224:
1221:
1200:
1199:
1184:
1180:
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1172:
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1163:
1160:
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1149:
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1135:
1132:
1127:
1122:
1119:
1116:
1112:
1108:
1105:
1102:
1099:
1096:
1063:) =
1030:
1029:
1016:
1012:
1008:
1005:
1002:
999:
996:
993:
990:
987:
981:
978:
971:
968:
965:
962:
959:
956:
953:
950:
947:
944:
941:
938:
935:
932:
926:
923:
905:
902:
881:
875:
874:
859:
856:
852:
848:
844:
840:
836:
832:
828:
824:
820:
817:
814:
811:
808:
803:
799:
795:
792:
789:
787:
785:
782:
779:
772:
769:
766:
760:
757:
753:
745:
740:
736:
732:
729:
727:
725:
722:
717:
713:
709:
706:
703:
698:
693:
690:
687:
683:
679:
678:
655:
631:
628:
625:
621:
590:
587:
584:
580:
557:
554:
550:
546:
542:
530:
529:
518:
515:
512:
506:
503:
499:
494:
488:
484:
480:
477:
474:
469:
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458:
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370:
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232:
229:
225:
200:, who applied
193:
190:
166:
165:
154:
151:
146:
142:
138:
135:
132:
127:
122:
119:
116:
112:
95:
92:
91:
46:
44:
37:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2215:
2204:
2201:
2200:
2198:
2183:
2175:
2174:
2171:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2136:
2134:
2132:
2128:
2122:
2119:
2117:
2114:
2112:
2109:
2107:
2104:
2102:
2099:
2097:
2094:
2092:
2089:
2087:
2084:
2082:
2081:Taylor series
2079:
2078:
2076:
2072:
2062:
2059:
2057:
2054:
2052:
2049:
2047:
2044:
2042:
2039:
2037:
2034:
2032:
2029:
2027:
2024:
2022:
2019:
2017:
2014:
2013:
2011:
2007:
2001:
1998:
1996:
1993:
1991:
1988:
1986:
1983:
1982:
1980:
1976:
1973:
1969:
1959:
1956:
1954:
1951:
1949:
1946:
1945:
1943:
1939:
1933:
1930:
1928:
1925:
1923:
1920:
1918:
1915:
1914:
1912:
1908:
1905:
1901:
1895:
1892:
1890:
1887:
1885:
1882:
1881:
1879:
1875:
1870:
1854:
1851:
1850:
1849:
1846:
1844:
1841:
1839:
1836:
1834:
1831:
1829:
1826:
1824:
1821:
1819:
1816:
1814:
1811:
1809:
1806:
1804:
1801:
1800:
1798:
1794:
1787:
1781:
1778:
1776:
1773:
1771:
1770:Powers of two
1768:
1766:
1763:
1761:
1758:
1756:
1755:Square number
1753:
1751:
1748:
1746:
1743:
1741:
1738:
1737:
1735:
1731:
1728:
1726:
1722:
1718:
1714:
1707:
1702:
1700:
1695:
1693:
1688:
1687:
1684:
1677:
1673:
1669:
1665:
1660:
1657:
1653:
1649:
1645:
1641:
1637:
1633:
1629:
1622:
1617:
1614:
1610:
1606:
1602:
1597:
1592:
1588:
1584:
1579:
1578:
1574:
1566:
1562:
1557:
1552:
1548:
1544:
1540:
1534:
1531:
1525:
1521:
1518:
1516:
1513:
1511:
1508:
1506:
1503:
1500:
1497:
1495:
1492:
1490:
1487:
1486:
1482:
1466:
1463:
1460:
1454:
1449:
1446:
1442:
1429:
1425:
1421:
1418:
1415:
1409:
1406:
1403:
1395:
1392:
1389:
1381:
1378:
1365:
1362:
1359:
1355:
1347:
1346:
1345:
1331:
1328:
1325:
1317:
1313:
1309:
1305:
1286:
1283:
1280:
1274:
1269:
1266:
1262:
1249:
1245:
1239:
1235:
1231:
1225:
1219:
1212:
1211:
1210:
1207:
1205:
1182:
1178:
1173:
1167:
1164:
1161:
1150:
1147:
1144:
1136:
1133:
1120:
1117:
1114:
1110:
1106:
1100:
1094:
1087:
1086:
1085:
1084:
1076:
1062:
1058:
1054:
1048:
1044:
1037:
1035:
1014:
1006:
1000:
997:
994:
988:
979:
976:
969:
963:
957:
954:
948:
942:
939:
933:
924:
921:
911:
910:
909:
903:
901:
899:
895:
891:
887:
880:
857:
854:
850:
846:
842:
838:
834:
830:
826:
822:
818:
815:
809:
801:
797:
793:
790:
788:
780:
777:
770:
767:
764:
758:
755:
751:
738:
734:
730:
728:
723:
720:
715:
707:
704:
691:
688:
685:
681:
669:
668:
667:
653:
629:
626:
623:
619:
608:
588:
585:
582:
578:
555:
552:
544:
516:
513:
510:
504:
501:
497:
492:
486:
478:
475:
462:
459:
456:
452:
447:
436:
432:
428:
425:
422:
417:
409:
406:
393:
390:
387:
383:
375:
374:
373:
356:
353:
350:
344:
341:
337:
331:
327:
316:
312:
306:
298:
295:
282:
279:
276:
272:
268:
265:
262:
257:
249:
246:
233:
230:
227:
223:
215:
214:
213:
211:
207:
203:
199:
191:
189:
187:
183:
179:
175:
171:
152:
149:
144:
136:
133:
120:
117:
114:
110:
102:
101:
100:
88:
85:
77:
67:
63:
57:
56:
50:
45:
36:
35:
30:
19:
2086:Power series
1828:Lucas number
1780:Powers of 10
1760:Cubic number
1667:
1663:
1631:
1627:
1586:
1582:
1546:
1533:
1315:
1311:
1301:
1208:
1203:
1201:
1074:
1060:
1056:
1052:
1046:
1042:
1038:
1033:
1031:
907:
885:
878:
876:
531:
371:
195:
167:
98:
80:
74:January 2021
71:
52:
1953:Conditional
1941:Convergence
1932:Telescoping
1917:Alternating
1833:Pell number
66:introducing
1978:Convergent
1922:Convergent
1556:1808.02841
1526:References
1501:(Grandi's)
206:factorials
178:factorials
49:references
2009:Divergent
1927:Divergent
1789:Advanced
1765:Factorial
1713:Sequences
1656:250892576
1591:CiteSeerX
1539:Euler, L.
1447:−
1435:∞
1426:∫
1407:−
1379:−
1371:∞
1356:∑
1344:, giving
1267:−
1255:∞
1246:∫
1165:−
1134:−
1126:∞
1111:∑
1049:) = (0,0)
998:−
980:˙
955:−
925:˙
894:Borel sum
888:) is the
858:…
816:≈
756:−
744:∞
735:∫
705:−
697:∞
682:∑
502:−
476:−
468:∞
453:∑
442:∞
433:∫
407:−
399:∞
384:∑
342:−
322:∞
313:∫
296:−
288:∞
273:∑
247:−
239:∞
224:∑
134:−
126:∞
111:∑
2197:Category
2182:Category
1948:Absolute
1541:(1760).
1483:See also
1202:Observe
1958:Uniform
1636:Bibcode
1613:2690371
1561:Bibcode
1079:
1065:
180:of the
62:improve
1910:Series
1717:series
1654:
1611:
1593:
877:where
605:. The
51:, but
1853:array
1733:Basic
1652:S2CID
1624:(PDF)
1609:JSTOR
1551:arXiv
819:0.596
198:Euler
174:Euler
168:is a
1793:list
1715:and
553:<
1672:doi
1644:doi
1601:doi
1051:as
855:369
851:499
847:078
843:341
839:074
835:194
831:323
827:362
823:347
609:of
2199::
1666:,
1650:,
1642:,
1632:62
1630:,
1626:,
1607:,
1599:,
1587:56
1585:,
1559:.
1036:.
900:.
188:.
1795:)
1791:(
1705:e
1698:t
1691:v
1674::
1668:3
1646::
1638::
1603::
1567:.
1563::
1553::
1467:.
1464:u
1461:d
1455:u
1450:u
1443:e
1430:1
1422:e
1419:=
1416:!
1413:)
1410:1
1404:n
1401:(
1396:1
1393:+
1390:n
1386:)
1382:1
1376:(
1366:1
1363:=
1360:n
1332:1
1329:=
1326:t
1316:t
1314:(
1312:x
1287:.
1284:u
1281:d
1275:u
1270:u
1263:e
1250:t
1240:t
1236:e
1232:=
1229:)
1226:t
1223:(
1220:x
1204:x
1183:n
1179:t
1174:!
1171:)
1168:1
1162:n
1159:(
1151:1
1148:+
1145:n
1141:)
1137:1
1131:(
1121:1
1118:=
1115:n
1107:=
1104:)
1101:t
1098:(
1095:x
1075:t
1071:/
1068:1
1061:t
1059:(
1057:y
1053:t
1047:y
1045:,
1043:x
1041:(
1034:t
1015:2
1011:)
1007:t
1004:(
1001:y
995:=
992:)
989:t
986:(
977:y
970:,
967:)
964:t
961:(
958:y
952:)
949:t
946:(
943:x
940:=
937:)
934:t
931:(
922:x
886:z
884:(
882:1
879:E
813:)
810:1
807:(
802:1
798:E
794:e
791:=
781:x
778:d
771:x
768:+
765:1
759:x
752:e
739:0
731:=
724:!
721:k
716:k
712:)
708:1
702:(
692:0
689:=
686:k
654:x
630:x
627:+
624:1
620:1
589:x
586:+
583:1
579:1
556:1
549:|
545:x
541:|
517:.
514:x
511:d
505:x
498:e
493:]
487:k
483:)
479:x
473:(
463:0
460:=
457:k
448:[
437:0
429:=
426:!
423:k
418:k
414:)
410:1
404:(
394:0
391:=
388:k
357:.
354:x
351:d
345:x
338:e
332:k
328:x
317:0
307:k
303:)
299:1
293:(
283:0
280:=
277:k
269:=
266:!
263:k
258:k
254:)
250:1
244:(
234:0
231:=
228:k
153:!
150:k
145:k
141:)
137:1
131:(
121:0
118:=
115:k
87:)
81:(
76:)
72:(
58:.
31:.
20:)
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