Question:

Test the series for convergence or divergence.

$\displaystyle

sn = \sum (-1)^n \frac{2n-1}{5n+1}

$

My attempt:

This is an alternating series, therefore, I will use the alternating series test.

We know that sn+1 < sn, however, the limit of sn is $\displaystyle \frac{2}{5}$ and not 0. Therefore, this series diverges.

Am I correct?

Edit: I tried the ration test: $\displaystyle \frac{sn+1}{sn}$ and it showed that the series is convergent because $\displaystyle \lim \frac{sn+1}{sn} = 0

$