There are quite a few simple results about convergent/divergent series derived from similar ones. So, here is something that I came across recently:
Let $\sum_{n=1}^\infty a_n$ consist of positive terms and be convergent. Define $b_n = \frac{1}{na_n^2}$. Is the series $\sum_{n=1}^\infty \frac{n}{\sum_{j=1}^n b_j}$ also convergent?
Some elementary manipulations on the terms of the derived series can bound its growth to $O(\log(n))$, but that's as far as I can get. Curious to know if the validity of this.
If this is too obvious or not appropriate for this forum, I will delete the question.
Best Answer
The answer is yes. Indeed, for integers $j\ge0$, let \begin{equation*} N_j:=\{2^j,\dots,2^{j+1}-1\},\quad A_j:=2^{-j}\sum_{k\in N_j}a_k, \end{equation*} so that $A_j$ is the arithmetic mean of the $a_n$'s over $n\in N_j$. Then \begin{equation*} A:=\sum_{j=0}^\infty 2^jA_j=\sum_{n\ge1}a_n<\infty \tag{1}\label{1} \end{equation*} and for $j\ge1$ and $n\in N_j$ \begin{equation*} \begin{aligned} s_n:=\sum_{k=1}^n b_k\ge\sum_{k\in N_{j-1}} b_k &\ge\frac12\, 2^{1-j}\sum_{k\in N_{j-1}} a_k^{-2} \\ &\ge\frac12\,\Big( 2^{1-j}\sum_{k\in N_{j-1}} a_k\Big)^{-2} =A_{j-1}^{-2}/2; \end{aligned} \tag{2}\label{2} \end{equation*} the latter inequality here is an instance of Jensen's inequality for the convex function $0<x\mapsto x^{-2}$.
Remark: Considering the equality case in this latter inequality suggests that the least favorable case is when the $a_n$'s with $n\in N_j$ differ insignificantly from one another (see the explanation "by \eqref{2}" in the multi-line display below) -- but then the $a_n$'s are relatively easy to deal with. Also, the factor $\frac1n$ in the definition $b_n:=\frac{1}{na_n^2}$ of $b_n$ varies insignificantly when $n\in N_j$ -- which was a reason to partition the set of all natural numbers into the blocks $N_j$. $\quad\Box$
Now we can write
\begin{equation*} \begin{alignedat}{2} \sum_{n\ge2}\frac n{s_n} &=\sum_{j\ge1}\sum_{n\in N_j}\frac n{s_n} & \\ &\le\sum_{j\ge1}2^j\frac{2^{j+1}}{A_{j-1}^{-2}/2} & \text{(by \eqref{2})} \\ &=\sum_{j\ge1}2^{2j+2}A_{j-1}^2 & \\ &\le \sum_{j\ge1}2^{2j+2}2^{-(j-1)}A\,A_{j-1} & \text{(by \eqref{1})} \\ &= 16A\sum_{j\ge1}2^{j-1}\,A_{j-1} =16A^2, & \end{alignedat} \end{equation*} so that the sum of the series in question is \begin{equation*} \frac1{s_1}+\sum_{n\ge2}\frac n{s_n}\le\frac1{s_1}+16A^2=a_1^2+16A^2<\infty. \quad\Box \end{equation*}