Convergence of the series $sumln(1+frac{(-1)^n}{n+1})$

I want to show that the series whose nth term is
$a_n=\ln(1+\frac{(-1)^n}{n+1})$ is convergent. I wanted to use the limit comparison test to compare it to the $p$ series but $a_n$ is not positive. I thought of writing the power series representation of $a_n$ using the power series representation of $\ln(1+x)$ with $x=b_n=\frac{(-1)^n}{n+1}$ we find that

$$a_n=b_n-\frac{1}{2}b_n^2+\frac{1}{3}b_n^3-\frac{1}{4}b_n^4+\cdots$$
Now the seris $\sum b_n$ is convergent by the alternating series test and the other terms are all terms of absolutely convergent series but it is an infinte sum, can I say so ? I mean is the infinite sum of convergent series a convergent series ? Is this correct and is there any other way to do it ?