Can we always know whether an infinite series converges or diverges, if we are summing over an elementary expression?

Question

Just like the question stated, can we always determine the convergence of an infinite sum of an elementary function?

For example, given some elementary function f(n) can we determine if the sum will converge or diverge for any elementary function?

$\sum_{n=0}^\infty f(n)$

Answer 1

Take $f(n) = \frac{1}{n^3 \sin^2(n)}$, which if you consider $\sin$ to be elementary is an elementary function.

The convergence of $\sum_{n=1}^\infty f(n)$, the Flint Hills series is an open problem.

Answer 2

First condition $f(n)$ must converge to $0$ for $n$ to $\infty$. 2. $f(n+1)/f(n) <1$ for $n$ to $\infty$ or root criterium. look up in wiki: https://en.wikipedia.org/wiki/Convergence_tests