<s>[INST] Proving that $\frac{1}{(2k+1)^4} - 2 \sum_{n=k+1}^\infty \left(\frac{1}{(2n)^4} - \frac{1}{(2n+1)^4}\right)$ is non negative Prove that the sequence
$$s_k = \frac{1}{(2k+1)^4} - 2 \sum_{n=k+1}^\infty \left(\frac{1}{(2n)^4} - \frac{1}{(2n+1)^4}\right)$$ is non-negative.
I would appreciate an elementary proof. I tried using series / integral comparison without success.
This question is a follow up of this one. There is an answer to that question that uses the integral representation of Hurwitz zeta function, that I'd like to avoid... if possible!
 [/INST] $f(x) = 1/x^4$ is strictly convex, so that
$$
 \frac{1}{(2n)^4} < \frac 12 \left(\frac{1}{(2n-1)^4} + \frac{1}{(2n+1)^4} \right) 
$$
or
$$
\frac{1}{(2n)^4} - \frac{1}{(2n+1)^4}  < \frac 12 \left(\frac{1}{(2n-1)^4} - \frac{1}{(2n+1)^4} \right) 
$$
It follows that
$$
2 \sum_{k=n+1}^\infty \left(\frac{1}{(2n)^4} - \frac{1}{(2n+1)^4} \right)
< \sum_{k=n+1}^\infty \left(\frac{1}{(2n-1)^4} - \frac{1}{(2n+1)^4} \right) = \frac{1}{(2k+1)^4} \, .
$$
</s>