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I'm generally interested in being able to find an asymptotic expansion of $$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$ As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...

Question. Numerically, the following is convincing. However, is there a proof? $$\left(\sum_{k\geq1}\frac1{\sqrt{2^k+3^k}}\right)^4 <\pi^2\left(\sum_{k\geq1}\frac1{2^k+3^k}\right)\left(\sum_{k\...