Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges

Question

Let $f$ be a non-negative decreasing function on $[1,+\infty)$. Prove that the series $$\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$$ converges.

score 9 · Answer 1 · answered Jul 07 '13 at 07:17

9

HINT:

Note that $$f(n+1) \leq \int_n^{n+1} f(x)dx \leq f(n)$$

answered Jul 07 '13 at 07:17

score 2 · Accepted Answer · answered Jul 07 '13 at 07:32

According to MathWorld, this is called MacLaurin-Cauchy Theorem. A proof can be found, for example, in Burkill's book A First Course in Mathematical Analysis. It is related to (generalized) Euler-Mascheroni constant. See also the proof in Wikipedia article on Integral test for convergence. The picture from this article (which I copied below) can help your intuition, when you try to prove this result.

Image copied from Wikipedia

To find more useful stuff you search for MacLaurin Cauchy theorem or MacLaurin Cauchy test, now that you know the name of the result.

Prove that the series $\sum_{n=1}^\infty \left[f(n)-\int_n^{n+1}\!f(x)\,\text{d}x\right]$ converges

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