The sum is extremely simple: $\sum_{n=1}^\infty \frac{1}{n^{2+(\cos{n})^2}}$.
The proof of convergence is elementary and is as follows:
Since $(\cos{n})^2 \geq 0$ for all positive $n$, we have $0<\frac{1}{n^{2+(\cos{n})^2}} \leq \frac{1}{n^2}$. Then by direct comparison, the sum converges.
This is trivial enough that I would expect a middle-schooler to be able to follow the argument, but somehow WA gets it wrong. Since WA is very powerful, I'm now somewhat suspicious about my proof, (although it seems pretty airtight). Is this a Wolfram bug or is my proof somehow flawed?
Here is the link to the sum in Wolfram: https://www.wolframalpha.com/input?i=sum+from+n%3D1+to+infinity+of+1%2F%28n%5E%282%2B%28cos%28n%29%29%5E2%29%29
Best Answer
Never mind Wolfram Alpha... You are correct, the series is convergent.
If you look more carefully to WolframAlpha's output, you'll see that "Standard computational time was exceeded". I believe this is the source of the (erroneous) conclusion.