The assignment:

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of real numbers and $f: [1, \infty) \rightarrow \mathbb{R}$ be a function, defined by $f(x) = a_n$, for $x \in [n,n+1).$ Show that: $$\lim_{b\to\infty} \int_1^b f(x) \ dx $$ exists if and only if $$\sum_{n=1}^\infty a_n$$converges. Similarly show that the equivalence also holds for the absolute improper integral and that if we have convergence the equality $\int_1^\infty f(x) \ dx = \sum_1^\infty a_n$ is true.

I think the comparison tests for both series and integrals might be helpful but I don't know which inequality to use to get from the series to the integral and vice versa, since I need either two integrals or two series to use a comparison test.

Any help would be appreciated.

  • $\begingroup$ See en.wikipedia.org/wiki/Integral_test_for_convergence#Proof. $\endgroup$ – user122283 Apr 30 '14 at 1:03
  • $\begingroup$ There shouldn't be a $\lim_{b\to\infty}$ in the first integral. $\endgroup$ – user122283 Apr 30 '14 at 1:04
  • $\begingroup$ @SanathDevalapurkar f is neither necessarily non-negative nor decreasing. $\endgroup$ – Nhat Apr 30 '14 at 1:06
  • $\begingroup$ Doesn't $f$ needs to be monotone? $\endgroup$ – Hawk Apr 30 '14 at 1:46

Hint: Note that $\displaystyle \int _1^\infty f=\sum \limits_{n=1}^\infty \left(\int _{n}^{n+1}f\right)=\sum \limits_{n=1}^\infty \left(a_n\right)$.

  • $\begingroup$ I like that hint, might just be what I was looking for. I'll try it. $\endgroup$ – Nhat Apr 30 '14 at 1:07
  • $\begingroup$ @KitKat Was it enough? $\endgroup$ – Git Gud Apr 30 '14 at 10:47
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    $\begingroup$ @KitKat I added something trivial, but perhaps you missed it. Does it help? $\endgroup$ – Git Gud Apr 30 '14 at 11:29
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    $\begingroup$ @KitKat Simply because $$\displaystyle \int _{n}^{n+1}f(x)\mathrm dx=\int _n^{n+1}f_{\mid [n, n+1[}(x)\mathrm dx=\int _n^{n+1}a_n\mathrm dx=a_n(n+1-n).$$ And please don't take my comment the wrong way. I said it because it's obvious you can handle this, you just happened to miss it. It happens to every one. $\endgroup$ – Git Gud Apr 30 '14 at 11:39
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    $\begingroup$ @KitKat Feel free to ask me any follow up questions you want. $\endgroup$ – Git Gud Apr 30 '14 at 12:14

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