Determining if a limit of a infinite series exists

Question

Let $f: (0,\infty) \to (0,\infty)$ be a continuous, decreasing, non-negative function.

I am trying to prove that a certain expression including this function has a limit.

Lets say I wanted to prove that the limit,

$$\lim_{n\to \infty} \sum_{k=1}^n f(k) $$

exists.

I think I want to use the Monotone Convergence Theorem to prove that the series converges, but for that I have to show that the sequence is bounded and here is were I get stuck.

Does the sum from 1 to infinity somehow make the infinite series bounded? (since it is decreasing?)

Is there another technique to make it bounded or am I looking at this problem from the wrong perspective?

I feel that I am a bit in over my head with this question, so any guidance would be greatly appreciated.

Answer 1

It depends on what you mean by the limit exists.

If you mean by that that you want it to be real, it is false, take

$$f(x)=1/x.$$

If you want the limit to be in $[0,+\infty]$, it exists since

$$\left(\sum_{k=0}^nf(k)\right)_n$$

is an increasing ($f$ is positive) positive sequence, so it must converges in $[0,+\infty]$.

Answer 2

Since $f$ assumes only non-negative values, the sequence $\sum_{k=1}^nf(k)$ is non-decreasing, therefore it has a limit in $[0,\infty]$.

Answer 3

Under your assumptions, the limit (that always exists, finite or $+\infty$) is finite if and only if the improper integral $$ \int_1^{+\infty} f(x)\, dx $$ is convergent. (This is the integral test for convergence of a series.)