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Unformatted text preview: BASIC SERIES AND CONVERGENCE TESTS Let ∑ ∞ n =1 a n be a series. Name Conditions Converges if Diverges if Test For lim | a n | = 0 conditions Divergence (TFD) satisfied Standard another series ∑ ∞ n =1 b n a n ≤ b n for all n b n ≤ a n for all n Comparison a n , b n ≥ 0 for all n ∑ b n conv ∑ b n div Test (SCT) Limit another series ∑ ∞ n =1 b n L < ∞ < L Comparison a n , b n ≥ 0 for all n ∑ b n conv ∑ b n div Test (LCT) lim n →∞ a n /b n = L Integral Test (IT) a n = f ( n ) ∞ 1 f ( x ) dx conv ∞ 1 f ( x ) dx div f continuous f positive f decreasing Alternating a n = (- 1) n +1 b n conditions Series Test (AST) b n positive satisfied lim n →∞ b n = 0 b n decreasing Ratio Test (RaT) lim n →∞ | a n +1 /a n | = L L < 1 L > 1 absolutely conv Root Test (RoT) lim n →∞ n | a n | = L L < 1 L > 1 absolutely conv Remarks : 1. Remember absolute value signs in ratio and root tests! They prove not just convergence but absolute convergence. 2. Check if a function is increasing/decreasing using derivatives, or using sum, product, composition, taking powers of increasing/decreasing functions. 3. If the test doesn’t work for the whole series, try removing the first few terms and test for ∑ ∞ n = N a n for some large enough N . BASIC SERIES Name Description Converges if Diverges if Geometric series ∞ n =1 ar n- 1- 1 < r < 1, sum is a 1- r otherwise p-series ∞ n =1 1 /n p p > 1 otherwise ESTIMATES AND BOUNDS Given a convergent series ∑ ∞ i =1 a i with sum s , define partial sum s n = ∑ n i =1 a i and remainder R n = s- s n . Given a Taylor series T ( x ) = ∑ ∞ i =0 c i ( x- a ) i for f ( x ) around x = a , define Taylor polynomial T n ( x ) = ∑ n i =0 c i ( x- a ) i and remainder R n ( x ) = T ( x )- T n ( x ). Name Estimate for R n Bound for s Remainder Estimate IF ∑ a n satisfies conditions IF ∑ a n satisfies conditions for Integral Test for the Integral Test, THEN for the Integral Test, THEN ∞ n +1 f ( x ) dx ≤ R n ≤ ∞ n f ( x ) dx s n + ∞ n +1 f ( x ) dx ≤ s ≤ s n + ∞ n f ( x ) dx Remainder Estimate IF ∑ b n convergent series for Std Comp Test ≤ a n ≤ b n for all n , THEN R n ≤ ∑ ∞ i =0 b i- ∑ n i =0 b i Remainder Estimate IF ∑ a n satisfies conditions IF ∑ a n satisfies conditions for Alternating Series for the Alt Series Test, THEN for the Alt Series Test, THEN | R n | ≤ b n s 2 n ≤ s ≤ s 2 n +1 Taylor’s Inequality Even if we do not know whether f ( x ) equals its Taylor series...
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This note was uploaded on 04/03/2010 for the course MATH 154 taught by Professor Hilliard during the Spring '08 term at Central Washington University.
- Spring '08