convergence of a series with logarithm

I've been dealing with the convergence of the following series for a while: $$\sum_{n=1}^\infty\ln\sqrt[n]{1+\frac{x}{n}}$$. I've tried the Cauchy test and found it reduces to study the convergence of $$\ln\left({1+\frac{x}{2^n}}\right)$$ But now I don't know how to study this one, hope someone may help. Thanks P.S According to the book it should converge for $x>-1$

Solution 1:

Note that Cauchy Condensation Test(Which you have applied) is valid for a non-increasing sequence of non-negative terms. So you should be careful as when $x<0$ the sequence is increasing .

Rather just use this :-

You can show by elementary calculus that $\ln(1+x)\leq x$. Also it is evident from the graph.

So $\frac{1}{n}\ln(1+\frac{x}{n})\leq \frac{x}{n^{2}}$.

As $\frac{1}{n^{2}}$ converges you have convergence for all $x>-1$.

And the reason for $x>-1$ is just that the series becomes undefined for $x\leq -1$ as $\log(1-x)$ becomes undefined(The term for $n=1$).