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?

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

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. — rushusuixing, Oct 18 '23 at 17:02
@rushusuixing A series of functions, such as summing over (1/3n) — Craftinators, Oct 18 '23 at 17:03
Your question is too broad. Do you mean a series or a series of function? if so what type of convergence? — julio_es_sui_glace, Oct 18 '23 at 17:03
@Craftinators Your example is not a series of functions...What I mean is somthing like $f(x)=\sum_{n=1]^\infty \frac{1}{n}sin(nx).$ — rushusuixing, Oct 18 '23 at 17:06
@rushusuixing I edited the question, hopefully its clearer now? — Craftinators, Oct 18 '23 at 17:08
It is a well-known result that the convergence of the series is consistent with $\int_a^\infty f(x) dx$. — rushusuixing, Oct 18 '23 at 17:11
There are some simple-seeming series whose convergence or divergence is unknown. See https://math.stackexchange.com/questions/20555/are-there-any-series-whose-convergence-is-unknown — Jair Taylor, Oct 18 '23 at 17:39

score 6 · Accepted Answer · answered Oct 18 '23 at 18:05

6

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.

answered Oct 18 '23 at 18:05

Theo Diamantakis

1

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. – Theo Diamantakis Oct 18 '23 at 18:07
Not rude, thank you for posting :) – Jair Taylor Oct 18 '23 at 18:28
Thank you very much for your help everyone – Craftinators Oct 18 '23 at 21:56

score 1 · Answer 2 · edited Oct 18 '23 at 23:34

1

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

edited Oct 18 '23 at 23:34

bob

answered Oct 18 '23 at 17:55

trula

2 Answers2