Question

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

Answer 1

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):}`

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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.

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I hope that this was helpful.