TITLE: About an equivalent condition for convergence of a scalar series
QUESTION [7 upvotes]: So I came across this problem a while back when one of my friends suggested it. The question asks to show that the scalar series $\sum_{n=1}^\infty a_n$ converges if and only if $$\sum_{n=1}^\infty \left(1-\frac1{2^2}+\cdots+\frac{(-1)^n}{n^2}\right)a_n$$ converges. The scalar field under consideration is real or complex. We were able to show the result holds if $a_n>0$. But the general case is eluding us.
Any ideas on how to approach this?

REPLY [5 votes]: The series has the form $\sum_n b_n a_n$ where $b_n \to b>0$ and $|b_n-b|<1/n^2$. Write $d_n=b_n-b$.
If $\sum_n a_n$ converges, then $|a_n|$ are bounded, so $\sum d_n a_n$ converges absolutely whence $\sum_n b_n a_n=\sum (ba_n + d_n a_n)$ also converges.
If $\sum_n b_n a_n$ converges, then the summands are bounded, so $|a_n|$ are also bounded.  Therefore $\sum d_n a_n$ converges absolutely whence
$$\sum ba_n =\sum (b_n a_n - d_n a_n)$$ also converges.