How do I determine if the alternating series sum_(n=1)^oo(-1)^n/sqrt(3n+1) is convergent?

1 Answer
Oct 18, 2014

Alternating Series Test

An 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.