Double sum with a lot of variables

Question

First, I want to thank everyone who is going to try. Here is the question. Show

$\sum_\limits{n\ge 1} \sum_\limits{m\ge n}m^d n^{c-d}\left(1+\frac{m^2}{rn}\right)^{-r}<\infty$. Here, $d>0$, $c>d$ and $r$ is very large. We can choose $r$ large enough if it helps.

This is what I thought at the beginning. For large enough $r$, there exists some constant $c_1$ such that

$\left(1+\frac{m^2}{rn}\right)^{-r}\le c_1 e^{-\frac{m^2}{n}}$.

Apparently, this is wrong because $\frac{m^2}{n}$ is increasing so I can't bound it uniformly.

Any help will be appreciated. Thanks in advance.

Answer 1

Hints: 1. For $r>0,$

$$\left(1+\dfrac{m^2}{rn}\right)^{-r} \le \left(\dfrac{m^2}{rn}\right)^{-r}.$$

2. For fixed $p<-1$ and $n=2,3,\dots,$

$$\sum_{m=n}^{\infty}m^p \le - \frac{(n-1)^{p+1}}{p+1},$$

this estimate coming from comparision with an integral.