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Ktu - s1 me module 1 (Calculus Module 1)

Anoop T Vilakkuvettom
Anoop T Vilakkuvettom

This document defines and explains several types of infinite series and tests used to determine if they converge or diverge. It introduces infinite series and sequences of partial sums. It then discusses tests for convergence/divergence including the geometric series test, limit comparison test, ratio test, root test, alternating series test, absolute convergence test, and radius of convergence for power series. Types of series covered include Taylor series and Maclaurin series expansions.

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Infinite Series 1 + 2 + 3 + =1 Sequence of Partial Sums 1 = 1 2 = 1 + 2 3 = 1 + 2 + 3 .. = 1 + 2 + + etc. 1 if the sequence of partial sums ( ) converges then the series =1 converges 2 lim is the sum of the series Geometric Series =1 = + + 2 + , ( 0) 1 Converges if || < 1 2 Diverges if || 1 Necessary condition for convergence 基 ＄ =1 ≠ lim = 0 lim 0 ≠ ≠ P Test lim 1 ｐ > 1, ｐ 1 Comparison Test If and If are two positive term series then if , then a) If is convergent then is also convergent b) If is divergent then is also divergent Limit form of comparison test: lim = 0 is a non-zero number. Then & converges or diverges together. DAlemberts Ratio Test If is a positive term series such that lim +1 = then the series a) Converges if < 1 b) Diverges if > 1 c) Test fails if = 1 Root Test If is a positive term series such that lim ( ) 1 = then the series a) Converges if < 1 b) Diverges if > 1 c) Test fails if = 1 Alternating series A series whose terms are alternatively positive and negative Leibnitz Test If the alternating series 1 2 + 3 is such that a) +1 , and b) lim = 0 Then the series converges Absolute convergence A series =1 is called absolutely convergent if | | =1 is convergent. If =1 is convergent and | | =1 is divergent we call the series conditionally convergent. Taylors Series The Taylors Series expansion of () about = is () = () + 1! () + ( )2 2! 霞() + ( )3 3! 霞霞() + .. Maclaurins Series Put = 0 in Taylors series () = (0) + ヰ(0) + 2 2! "(0) + 3 3! 霞霞(0) + . Radius of Convergence If =1 is a power series which converges absolutely for || < and those absolute term diverges for || > and || = if either converges or diverges absolutely. Then R is knowns as the Radius of convergence of the power series.

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Ktu - s1 me module 1 (Calculus Module 1)

  • 1. Infinite Series 1 + 2 + 3 + =1 Sequence of Partial Sums 1 = 1 2 = 1 + 2 3 = 1 + 2 + 3 .. = 1 + 2 + + etc. 1 if the sequence of partial sums ( ) converges then the series =1 converges 2 lim is the sum of the series Geometric Series =1 = + + 2 + , ( 0) 1 Converges if || < 1 2 Diverges if || 1 Necessary condition for convergence 基 ＄ =1 ≠ lim = 0 lim 0 ≠ ≠ P Test lim 1 ｐ > 1, ｐ 1 Comparison Test If and If are two positive term series then if , then a) If is convergent then is also convergent b) If is divergent then is also divergent Limit form of comparison test: lim = 0 is a non-zero number. Then & converges or diverges together. DAlemberts Ratio Test If is a positive term series such that lim +1 = then the series a) Converges if < 1 b) Diverges if > 1 c) Test fails if = 1 Root Test If is a positive term series such that lim ( ) 1 = then the series a) Converges if < 1 b) Diverges if > 1 c) Test fails if = 1 Alternating series A series whose terms are alternatively positive and negative Leibnitz Test If the alternating series 1 2 + 3 is such that a) +1 , and b) lim = 0 Then the series converges Absolute convergence A series =1 is called absolutely convergent if | | =1 is convergent. If =1 is convergent and | | =1 is divergent we call the series conditionally convergent. Taylors Series The Taylors Series expansion of () about = is () = () + 1! () + ( )2 2! 霞() + ( )3 3! 霞霞() + .. Maclaurins Series Put = 0 in Taylors series () = (0) + ヰ(0) + 2 2! "(0) + 3 3! 霞霞(0) + . Radius of Convergence If =1 is a power series which converges absolutely for || < and those absolute term diverges for || > and || = if either converges or diverges absolutely. Then R is knowns as the Radius of convergence of the power series.