# 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.