Series Convergence using Cauchy's test

Question

Trying to figure out if the following series converence:

$$\sum_{n=1}^{\infty} \frac{(n+1)}{n^{n/2}}-\frac{n^{1/2}}{(n+1)}$$

I tried using Cauchy's test but it seems too complecated.

I'd appriciate if anyone has any idea!

Answer 1

so to me it seem that $$ \sum_{n=1}^{\infty} \frac{n+1}{n^\frac{n}{2}}-\frac{n^\frac{1}{2}}{n+1} \le \sum_{n=1}^{\infty} \frac{1}{n+1}- \frac{n^\frac{1}{2}}{n+1} \le \sum_{n=1}^{\infty} - \frac{n^\frac{1}{2}}{2(n+1)} \le \sum_{n=1}^{\infty} - \frac{n^\frac{1}{2}}{3n} $$ its since its clearly diverge to $ -\infty $ so dose to original series, i could explain why its diverge if you need me to but its quit easy