Proving convergence of series $\sum_{n=1}^{\infty}u_{n}$ I was asked to prove that the series; $\sum_{n=1}^{\infty}u_{n}$ converges, given that,

$\lim_{n \rightarrow \infty}n^{p}u_{n}=A&lt;\infty$ and $p&gt;1$.

The proof of this is:

If $u_{n}&lt;\frac{A}{n^{p}}$ then applying the integral test shows convergence.

This may be a stupid question but why is it that we are able to make the assumption $u_{n}&lt;\frac{A}{n^{p}}$ ?

Thanks.
Besides direct comparison, you have $ u_n \sim A / n^p $ hence $\sum u_n$ converges by limit comparison with $\sum A/ n^p.$
Besides direct comparison, you have $ u_n \sim A / n^p $ hence $\sum u_n$ converges by limit comparison with $\sum A/ n^p.$