Convergent or divergent?

[much2know]

1/2-2/9+3/28-4/65+...

I do not know which test to use to determine if the series is convergent (comparison, ratio,.ect..). Maybe it is easier than I think.

 

[tkhunny]

Decreasing terms.
Alternating.

Are we done?

Truthfully, with only four terms, there just isn't enough information.

 

[soroban]

Hello, much2know!

tkhunny is right . . . Did you understand his hint?

\(\displaystyle \displaystyle \frac{1}{2} - \frac{2}{9} + \frac{3}{28} - \frac{4}{65} + \hdots\)

I do not know which test to use to determine if the series is convergent
(comparison, ratio, etc.). .Maybe it is easier than I think.

Alternating series: .\(\displaystyle a_o - a_1 + a_2 - a_3 + \hdots\)

If \(\displaystyle a_n\,>\,a_{n+1}\,\text{ (the terms are decreasing),\,the series is convergent.}\)

The series seems to be: .\(\displaystyle \displaystyle \sum^{\infty}_{n=0}(\text{-}1)^n\frac{n}{n^3+1} \)

So what do you think?

 

[much2know]

Thankyou! The series is convergent.