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

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?

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

• Is it a series of numbers or functions? As for funtions, there are differences between point-to-point convergence and uniform convergence, both of which are easy to determine. Oct 18, 2023 at 17:02
• @rushusuixing A series of functions, such as summing over (1/3n) Oct 18, 2023 at 17:03
• Your question is too broad. Do you mean a series or a series of function? if so what type of convergence? Oct 18, 2023 at 17:03
• @Craftinators Your example is not a series of functions...What I mean is somthing like $f(x)=\sum_{n=1]^\infty \frac{1}{n}sin(nx).$ Oct 18, 2023 at 17:06
• @rushusuixing That's only the case for monotone $f$. Oct 18, 2023 at 17:49

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.

• I see that Jair Taylor beat me to it in the comments without me realising. Hopefully posting a similar answer to his comment isn't considered rude. Oct 18, 2023 at 18:07
• Not rude, thank you for posting :) Oct 18, 2023 at 18:28
• Thank you very much for your help everyone Oct 18, 2023 at 21:56

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