Problem: For real numbers $x\ge1$ and $k>0$, let $f:R\rightarrow R$ and $g:R\rightarrow R$ be defined as follows.
$f(x) = -\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$ ,
$g(x)=\frac{1}{2x^2}-\frac{1}{x}+\sum_{n=1}^{\infty}\frac{1}{(nk+x)^2}$.
Show that $f$ is increasing and $g$ is decreasing (that is, $f'(x)\ge0$ and $g'(x)\le0$ ).
I was given this problem to solve. I tried differentiating each function with respect to $x$ and got the following:
$f'(x) = \frac{1}{x^2}-2\sum_{n=1}^{\infty}\frac{1}{(nk+x)^3}$ ,
$g'(x)=-\frac{1}{x^3}+\frac{1}{x^2}-2\sum_{n=1}^{\infty}\frac{1}{(nk+x)^3}$.
I was unable to proceed after this stage. Also i tried the following.
For $x>y\ge1$, i wanted to show that $f(x)-f(y)\ge0$, however, i got
$f(x)-f(y)=\frac{x-y}{xy}+\sum_{n=1}^{\infty}\left[\frac{1}{(nk+x)^2}-\frac{1}{(nk+y)^2}\right]$. I got a similar expression for $g$. Once again, I am unable to proceed from here.
Your solution(s) to this problem will greatly help me. Thank you in advance.