calculus and analysis – How to judge whether the series $sum_{n=1}^{infty} (-1)^{n} ln left(1+frac{1}{sqrt{n}}right)$ converges


Given $u_{n}=(-1)^{n} ln left(1+frac{1}{sqrt{n}}right)$, I want to judge the convergence of series $sum_{n=1}^{infty} u_{n}$ and series $sum_{n=1}^{infty} u_{n}^{2}$.

SumConvergence((-1)^n Log(1 + 1/Sqrt(n)), n, Method -> "RatioTest")
SumConvergence((-1)^n Log(1 + 1/Sqrt(n)), n, Method -> "RootTest")
SumConvergence((-1)^n Log(1 + 1/Sqrt(n)), n, Method -> "RaabeTest")
SumConvergence((-1)^n Log(1 + 1/Sqrt(n)), n, Method -> "IntegralTest")

But the above four methods can not judge whether the series $sum_{n=1}^{infty} u_{n}$ converges or not.

What method should I use to quickly judge whether the above series converges?

The answer is that series $sum_{n=1}^{infty} u_{n}$ converges and series $sum_{n=1}^{infty} u_{n}^{2}$ diverges.