convergence_tests

convergence_tests - Summary of Convergence Tests for Series...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Summary of Convergence Tests for Series Test term test (or the zero test) n th X I 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 1 x 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 1 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 in nite 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 in nite X j nj a X I X n X (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 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online