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Unformatted text preview: Summary of Convergence Tests for Series
Test term test (or the zero test)
n th X
n n Series
an Convergence or Divergence Diverges if lim
n n Comments Inconclusive if lim
n Geometric series =0 ax or I X
n =1 ax 1 ! 3I an T= 0
a 3I an = 0. Converges to only if jxj < 1 Diverges if jxj ! 1 Useful for comparison tests if the nth term an of a series is similar to axn . Useful for comparison tests if the nth term an of 1 a series is similar to p .
n p -series I X
n Converges if p > 1 Diverges if p Converges if Diverges if
c ZI c =1 p 1 Integral an (c ! 0) = an = f (n) for all n I X ZI f x ( ) dx converges n c f x ( ) dx diverges X Comparison an and X bn X X X X bn converges =A diverges =A X with 0 X an bn an and X for all n
bn an X an converges bn diverges converges Limit Comparison* with an ; bn > 0 for all n and lim
n bn converges =A diverges =A X 3I bn
n an =L>0 bn X an an diverges The function f obtained from an = f (n) must be continuous, positive, decreasing and readily integrable for x ! c. The comparison series X bn is often a geometric series or a p-series. The comparison series X bn is often a geometric series or a p-series. To nd bn consider only the terms of an that have the greatest eect on the magnitude. Inconclusive if L = 1. Useful if an involves factorials or nth powers. Test is inconclusive if L = 1. Useful if an involves th n powers. Useful for series containing both positive and negative terms. Applicable only to series with alternating terms. Ratio X X an with lim 3I j an j nj
a a +1 j = L Converges (absolutely) if L < 1 Diverges if L > 1 or if L is innite Converges (absolutely) if L < 1 Root* Absolute Value an with lim p n j j= n n3I
an L Diverges if L > 1 or if L is innite X j nj
(an > 0) j n j converges =A
a X an converges Alternating series =1 ( 1)n 1 an Converges if 0 < an+1 < an for all n and lim an = 0
n 3I *The Root and Limit Comparison tests are not included in the current textbook used in Calculus classes at Bates College. 1 ...
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