$ |a_{n+1}/a_n| \leq n^2/(n+1)^2 $ for all natual numbers $n$, prove that the series formed by $a_n$ converges absolutely

Question

$ \left|\dfrac{a_{n+1}}{a_n}\right| \leq \dfrac{n^2}{(n+1)^2} $

We are given the above infinite series and are to prove its absolute convergence. Clearly, ratio test isn't of use here and I can't come up with any thing else.

Maybe it's a tough one

Answer 1

We may assume that $a_n>0$ for all $n\geq1$. It then follows that $$(n+1)^2 a_{n+1}\leq n^2 a_n\qquad(n\geq1)\ ,$$ so that induction implies $a_n\leq{a_1\over n^2}$. Therefore the series $\sum_{n\geq1} a_n$ converges.