I would like an example of a function that is continuous and positive and has the following properties: $$\int_a^{\infty}f(x) dx $$ is convergent and $$\lim_{x \to \infty} f(x) \not = 0$$ (I think the limit should not exist).
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1Take spikes of height $1$ centered at the integers whose bases have lengths tending to zero sufficiently fast. Add $e^{-x}$ to make things positive. – David Mitra Mar 21 '14 at 22:13
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Thank you for your answer. That was my idea as well, but there is no way the function can be described without "branches", right? – user42768 Mar 21 '14 at 22:22
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You could express the "spikes" as an infinite sum of compressions and translations of a "basic spike". – David Mitra Mar 21 '14 at 22:25
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And how would the definition of the function look like in that case? – user42768 Mar 21 '14 at 22:29
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1Let $g$ be piecewise linear with support on $[-1,1]$, increasing on $[-1,0]$, decreasing on $[0,1]$, and with $g(0)=1$. Then take as the spike function $h(x)=\sum_{n=1}^\infty g\bigl(n^2(x-n)\bigr)$. – David Mitra Mar 21 '14 at 22:33
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Very nice. Thank you! – user42768 Mar 21 '14 at 22:41
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when the limit can be zero? Any idea? – Topology Dec 19 '14 at 18:17
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This (more recent) post answers your question: If the integral of a positive/negative function is convergent, then the limit of the function will be zero? – Anne Bauval May 15 '23 at 11:22