Is Riemann Sum increasing if $f$ is continuous, decreasing and concave?

Asked Apr 25 '16 at 18:07

Active Apr 25 '16 at 18:34

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$$f: [0,1] \rightarrow \Bbb{R} \;\; \text{(the set of real numbers).}$$ $$f(0) = 1\;, \;\;f(1) = 0$$ $$f \;\text{is continuous, decreasing and concave.}$$ $$ S_n = \sum_{i=1}^n f\left(\frac{i}{n}\right)\frac{1}{n}.$$

Question: Is the sequence $S_n$ increasing?

edited Apr 25 '16 at 18:19

asked Apr 25 '16 at 18:07

acamath 72

Under the (further) assumption that $f$ is $C^2$, have you tried doing an asymptotic development of $S_n$ to order $1/n^2$, in a fashion similar to that of this question? – Clement C. Apr 25 '16 at 18:51
Thanks for comment, but I don't know what 'asymptotic development of $S_n$' mean. Is it such like Taylor Expansion? – acamath 72 Apr 26 '16 at 15:17
Yes -- see the link above. – Clement C. Apr 26 '16 at 15:18
Oh, I realize the link now.... Thank you!! – acamath 72 Apr 26 '16 at 15:32

Is Riemann Sum increasing if $f$ is continuous, decreasing and concave?

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