I need to find an example of a positive function f, and a constant a>0 such that the improper integral of f from 0 to infinity converge, but the series f(na) from 1 to infinity diverge.
2026-05-15 15:34:37.1778859277
Find an example of a positive function that its improper integral converge, but its series diverge
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To expand on Jyrki's comment, let $$f(x)=\begin{cases} 1 &\text{ if there exists an }n\in\mathbb{N}\text{ with } |x-na|\leq \frac{1}{n^2}\\ 0 &\text{ otherwise} \end{cases}$$ The graph looks like a sequence of thinner and thinner rectangles above the points $na$.
(1) What is $\sum_{n=0}^\infty f(na)$?
(2) Show that $\int_0^\infty f(x)\ dx$ is equal to a convergent series.