Assume I am studying the series of the form $$\sum_{k=1}^\infty f(k)$$ where we assume $f$ to be some continuous monotonic function over the interval $\langle1,+\infty)$. We know that for these function we have this bound: $$f(1)+\int_1^nf(x)dx\leq\sum_{k=1}^nf(k)\leq f(n)+\int_1^nf(x)dx$$ or $$f(n)+\int_1^nf(x)dx\leq\sum_{k=1}^nf(k)\leq f(1)+\int_1^nf(x)dx$$ if $f$ is decreasing. My goal is to determine the behaviour of the partial sum $s_n=\sum_{k=1}^nf(k)$. More precisely to find sequence $a_n$ such that $s_n\sim a_n$ in the sense that $\lim_{n\to\infty}\frac{a_n}{s_n}=1.$My question is: Can i explicitly say that $$s_n\sim\int_1^nf(k)$$ Assume there exists primitive $F$ to $f$ then rewriting the upper inequality (for increasing functions) $$f(1)+F(n)-F(1)\leq s_n \leq f(n)+F(n)-F(1)$$
It seems that this should always hold but I can't seem to relate functions with their primitives. In fact $f(1), F(1)$ are some constants and $f(n)$ vanishes compared to $F(n)$ so $s_n$ should be bounded by $F(n)$ thus $s_n\sim F(n)$. Is that true? If $f$ is a function and $F$ its primitive, what can we say about the limit $$\lim_{n\to\infty}\frac{f(n)}{F(n)}$$ What about $$f(n)\in O(F(n))$$ Does it hold?
If $f$ increases very rapidly, then it's not true that $f(n)$ is negligible compared to $F(n)$. For instance, $f(x)=e^x$, then $F(n)=f(n)=e^n$