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I know of several convergence/ divergence tests for infinite sums, but strangely the only tests I know for improper integrals (which seemingly are just continuous versions of infinite sums) are the comparison test and the p-test. Can we apply all of the same tests we use to determine convergence of infinite sums to improper integrals? If not, why?

In particular, I'm looking at an annoyingly difficult improper integral at the moment and I'm noticing that if the $n$th term test applies to improper integrals as well as infinite series, it'll diverge. So can I apply this test? If not, why?

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  • $\begingroup$ If the function is positive and decreasing, the ideas for sums should transfer. Cannot say more without seeing the problem. $\endgroup$ – André Nicolas Mar 15 '15 at 3:29
  • $\begingroup$ That's fine -- I've figured out another way to do it (it does diverge). I really just want to know when I can apply tests for convergence for infinite series to improper integrals. What conditions do integrals need to be able to apply: the $n$th term test, the limit comparison test, the ratio test, the root test, etc? $\endgroup$ – user223699 Mar 15 '15 at 3:32
  • $\begingroup$ What conditions we need may be a complicated problem. But for $\int_0^\infty$, $f$ continuous, bounded at the left end, positive non-increasing should be sufficient for all the tests you mentioned. $\endgroup$ – André Nicolas Mar 15 '15 at 3:40
  • $\begingroup$ OK. Good. Thanks. :) $\endgroup$ – user223699 Mar 15 '15 at 3:41
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As was mentioned in the comments, here is a test that may help you formalize using techniques normally reserved for series:

If $f:[1,\infty)\mapsto[0,\infty)$ is non-increasing and is Riemann integrable on $[1,b]$ for any $b>1$, then $\int_1^\infty f(x)dx$ converges if and only if $\sum_{n=1}^\infty f(n)$ converges.

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