Consider the following sequence $a_n=\dfrac{\ln n}{n^p}$ where $p>1$. I want to check for the convergence of the series of that sequence i.e. $\displaystyle \sum_{n=1}^{\infty}a_n$.
I intuitively feel that the series is indeed convergent. Here is my idea - Since $p>1$ there must be some $\epsilon$ such that $p>\epsilon>1$. Then, $a_n=\dfrac{\ln n}{n^\epsilon \cdot n^{p-\epsilon}}$ and since polynomial grows faster than logarithm, there must be some $N_0\in \mathbb{N}$ such that $\ln n<n^{p-\epsilon}$ $\forall$ $N_0<n\in\mathbb{N}$. Then for all $n>N_0$, $a_n<\dfrac{1}{n^\epsilon}$, and since the series of the sequence $\dfrac{1}{n^p}$ converges for all $p>1$, $a_n$ must also converge (as $\epsilon>1$).
If the above proof is correct, I would like to know a more formal proof (using comparison tests or limit tests or fundamental properties like cauchy criterion/monotone convergence theorem). If the above proof is incorrect, Please tell me where I went wrong.
