Determining if series converges or diverges

Question

The Series is

For this series the ratio test is inconclusive. I have rewritten the series as

Currently i am approaching the problem using limit test.

I couldn't progress from this point. Any help or hints are greatly appreciated! Thanks!

Answer 1

Hint:

I think you could use Raabe's test.

Answer 2

The terms decrease each step by a factor $2n/(2n+3)$.
So their logarithm decreases by $\frac{3}{2n}+O(n^{-2})$.
The logarithm of the $n^{th}$ term is around $-\frac{3}{2}\log n$, and the term itself is around $n^{-3/2}$.
I think the original sum ought to converge.