Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$

Question

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$$

Suppose $\displaystyle\sum^\infty_{n=1}a_n$ does converge, so by the comparsion test the given series also converge.

Suppose $\displaystyle\sum^\infty_{n=1}a_n$ does not converge:

• If $a_n$ is a bounded sequence with a bound $M$ then:

  $\forall n \ a_n\le M \Rightarrow \large\frac{a_n}{1+n^2a_n}>\frac{a_n}{1+M}\to\infty$

  So the given series diverge.

• If $a_n$ isn't bounded, it has a subsequence that tends to infinity, so we have: $\displaystyle\frac{a_{n_k}}{1+{n_k}^2a_{n_k}}\longrightarrow^{k\to\infty}\infty$ so the given series will diverge.

(Couldn't find the tex for the limit with arrow)

Answer 1

Expanding on the other answer because it was not immediately obvious to me, if $a_n = 0$, you can discard that term because it's $0$. So assume wlog that $a_n \neq 0$.

Then $\dfrac{a_n}{1 + a_nn^2} = \dfrac{1}{\frac{1}{a_n} + n^2} < \dfrac{1}{n^2}$, which gives you convergence always.

Answer 2

Hint: $\dfrac{a_n}{1 + n^2a_n} \leq \dfrac{1}{n^2}$