Summary of Convergence Tests for Series Test term test (or the zero test) n Series X th Geometric series 1 X n =0 or n ax -series n =1 n 1 X 1 X =1 n ax 1 ! Comparison an and n Root* X X an !1 bn with lim an with lim X bn a 1 X !1 X bn n =1 j an X +1 j = L j nj a p n j j= n n!1 a Z1 Z 1c ( ) dx converges f x ( ) dx diverges bn converges =) an diverges =) bn converges =) bn diverges =) X X bn X X an diverges an an converges converges diverges Converges (absolutely) if L < 1 Diverges if L > 1 or if L is in nite Diverges if L > 1 or if L is in nite X (an > 0) j n j converges =) a X an converges Converges if 0 < an+1 < an for all n and lim an = 0 n !1 Useful for comparison tests if the nth term an of a series is similar to axn . n f x c !1 an = 0. Useful for comparison tests if the nth term an of 1 a series is similar to p . Converges (absolutely) if L < 1 L an ( 1)n 1 an X =L>0 n X j nj Alternating series X an Absolute Value X Diverges if with an ; bn > 0 for all n and lim Ratio Converges if with 0 an bn for all n X Limit Comparison* X and only if jxj < 1 Diverges if p 1 p an (c 0) = an = f (n) for all n an x n Converges if p > 1 n c X a 1 Inconclusive if lim Diverges if jxj 1 1 X Integral !1 an 6= 0 Converges to 1 n Comments Diverges if lim an n p Convergence or Divergence The function f obtained from an = f (n) must be continuous, positive, decreasing and readily integrable for x c. The X comparison series bn is often a geometric series or a p-series. The X comparison series bn is often a geometric series or a p-series. To nd bn consider only the terms of an that have the greatest e ect 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. *The Root and Limit Comparison tests are not included in the current textbook used in Calculus classes at Bates College. 1