# analysis – a guess relation betwenn series and infinite integral convergence

Background: Our real analysis textbook says if function f is a non-negative decreasing function on range $$(1,+infty)$$, thus $$int_{1}^{+infty}f(x),dx$$ converges is equivalent to $$sum_{n=1}^{infty}f(n)$$ converges.

So I have thought about other properties f(x) may have such that infinite integral and sum have the same convergece like that above. I guess uniform continuity is a good candidate, and I’ve tried to find some functions such that this doesn’t hold but failed.

So If this is really true?Or for f(x) with some properties it doesn’t holds?