Geometric Series ar n n r 1 r 1 Sum S a 1 r Telescoping b n b n 1 n 1 lim n b n

Geometric series ar n n r 1 r 1 sum s a 1 r

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Geometric Series    arnn0  r1r1     Sum: Sa1rTelescoping bnbn1n1      limnbnLSum: Sb1Lp-Series 1npn1      p1p1Alternating Series 1n1ann1  0an1anand  limnan0Remainder:RNaN1Integral( f  is continuous,positive, anddecreasing)ann1,anf n 0fx dx converges1  fx dx diverges1Remainder:0RNfx dxNRoot ann1limnann1limnann1Test is inconclusive iflimnann1.Ratio ann1limnan1an1limnan1an1Test is inconclusive iflimnan1an1.Direct comparisonan, bn0  ann10anbn andbnn1 converges0bnan  and bnn1 divergesLimit Comparisonan, bn0ann1limnanbnL0andbnn1 convergeslimnanbnL0andbnn1 diverges
21                                       Power Series for Elementary FunctionsFunction                                                                                                                          Interval of Convergence  1x     1x1x12x13x141nx1n                                0x211x  1xx2x3x4x51nxn                                                              1x1ln x    x1x122x133x1441n1x1nn                                 0x2ex      1xx22!x33!x44!x55!xnn!                                                                   xsin x  xx33!

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