real analysis - Convergence of the series $sumlimits_{n=1}^{infty}frac1nlogleft(1+frac1nright)$.

I am trying to check the convergence or divergence of the series $\displaystyle\sum_{n=1}^{\infty}\dfrac1n\log\left(1+\dfrac1n\right)$.

My attempt: for a finite
$p$,

$$\begin{align}\displaystyle\sum_{k=n}^{n+p}\dfrac1k\log\left(1+\dfrac1k\right)&\lt\dfrac1n\displaystyle\sum_{k=n}^{n+p}\log\left(1+\dfrac1k\right)\\&=\dfrac1n\log\large\Pi_{k=n}^{n+p}\left(\dfrac{k+1}{k}\right)\\&=\dfrac1n\log\left(1+\dfrac{p+1}{n}\right)\\&\lt\dfrac1n\log2,\text{ for large $n$ and $p$ is finite.}\\&\lt\varepsilon\end{align}$$
Hence the series converges.

Answer

Because

$$\frac{\frac{1}{n}\ln(1+\frac{1}{n})}{\frac{1}{n^2}}\rightarrow1$$ and
$$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}.$$