# Thread: Problems proving a summation

1. ## Problems proving a summation

I have been trying for a week to prove a result from a textbook and have gotten nowhere. Please help.

I have the facts $q^{-\delta}$ $logq$ $\rightarrow$ $\infty$ for each $\delta>0$ and I'm trying to prove for each $\epsilon>0$
$\sum_{q=1}^{\infty}((logq)/(q^{1+\epsilon}))<\infty$

2. ## Re: Problems proving a summation

Originally Posted by klw289
I have been trying for a week to prove a result from a textbook and have gotten nowhere. Please help.

I have the facts $q^{-\delta}$ $logq$ $\rightarrow$ $\infty$ for each $\delta>0$ and I'm trying to prove for each $\epsilon>0$
$\sum_{q=1}^{\infty}((logq)/(q^{1+\epsilon}))<\infty$
This is series of tricks. I am going to use easier notation.
Suppose that $p>1$ and let $r=\frac{p-1}{2}$.

Set some things up first. $\log(n)=r^{-1}\log(n^r)\le r^{-1}n^r$

So $\dfrac{\log(n)}{n^p}\le\dfrac{n^r}{r\cdot n^p}=\dfrac{1}{r\cdot n^{p-r}}$

If we note that $p-r=\frac{p+1}{2}>1$.

So you have a p-series which converges.