Alternating Series Test (Leibniz's Theorem) for Convergence of an Infinite Series - Calculus | Socratic

Alternating Series TestAn alternating series #sum_{n=1}^infty(-1)^n b_n#, #b_n ge 0# converges if both of the following conditions hold.#{(b_n ge b_{n+1} " for all " n ge N),(lim_{n to infty}b_n=0):}#Let us look at the posted alternating series.In this series, #b_n=1/sqrt{3n+1}#.#b_n=1/sqrt{3n+1} ge 1/sqrt{3(n+1)+1}=b_{n+1}# for all #n ge 1#.and#lim_{n to infty}b_n=lim_{n to infty}1/sqrt{3n+1}=1/infty=0#Hence, we conclude that the series converges by Alternating Series Test.I hope that this was helpful.