Summary of Convergence and Divergence Tests for Series Test Divergence 1 P Series Convergence or Divergence Diverges if lim an 6= 0 an Geometric 1 P p-series ar Integral (2) Diverges if |r| > 1 an if the nth -term an of a series is similar to arn . if the nth -term an of a series (2) Diverges if p 6 1 is similar to 1 P bn converges and an 6 bn for every n, then 1 P bn n=1 an > 0, bn > 0 (2) If 1 P n=1 1 P 1 . np The function f obtained (1) Converges if 1 f (x) dx converges; R1 (2) Diverges if 1 f (x) dx diverges n=1 n=1 if |r| < 1; (1) Converges if p > 1; (1) If an , r R1 an = f (n) 1 P 1 Useful for comparison tests n=1 Comparison a (1) Converges to S = 1 1 P p n=1 n 1 P n!1 Useful for comparison tests n n=0 series Inconclusive if lim an = 0 n!1 n=1 Comments an converges; n=1 bn diverges and an > bn 1 P for every n, then an diverges; n=1 an (3) If lim > 0 (not 1), then n!1 bn both series converge or both diverge. from an = f (n) must be continuous, positive, decreasing, and readily integrable. The comparison series 1 P bn is often a geometric n=1 series or a p-series. To find bn in (3), consider only the terms of an that have the greatest e↵ect on the magnitude. Inconclusive if L = 1. Useful Ratio 1 P an n=1 an+1 If lim = L (or 1), the series n!1 an if an involves factorials or (1) converges (absolutely) if L < 1; nth powers. If an > 0 for (2) diverges if L > 1 (or 1) every n, the absolute value sign may be disregarded. If lim Root 1 P n!1 an n=1 Alternating 1 P ( 1)n an n=1 Series an > 0 p n |an | = L (or 1), the series Inconclusive if L = 1. Useful if an involves nth powers. (1) converges (absolutely) if L < 1; If an > 0 for every n, the (2) diverges if L > 1 (or 1) absolute value sign may be disregarded. Converges if ak > ak+1 for every k Applicable only to an and lim an = 0. alternating series. n!1