For any $f\in C^1[0,1]$ show that $\sum_{k=1}^n f\big(\frac{k}{n}\big) - n \int_0^1 f(x)dx$ converges to $\frac{f(1)-f(0)}{2}$ as $n \to \infty$.

Asked Jul 06 '22 at 17:51

Active Jul 06 '22 at 21:44

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For any $f\in C^1[0,1]$ show that $$ \sum_{k=1}^n f\big(\frac{k}{n}\big) - n \int_0^1 f(x)dx$$ converges to $$\frac{f(1)-f(0)}{2}$$ as $n \to \infty$.

I'm completely completely stuck here. Don't know where to start. Some hint would be really appreciated. Can anyone just give me some hints please? I'm not asking for a full solution, just some guidance so that I could do it on my own too.

asked Jul 06 '22 at 17:51

Itachi

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https://math.stackexchange.com/q/914240/42969, https://math.stackexchange.com/q/2303994/42969, https://math.stackexchange.com/q/1472926/42969 – all found with Approach0 – Martin R Jul 06 '22 at 17:56
@MartinR Thank you very much. – Itachi Jul 06 '22 at 17:58
Also recently asked here: https://math.stackexchange.com/q/4487454/42969 – Martin R Jul 06 '22 at 17:58
@MartinR I didn't know about "approach 0". So is this a search engine for maths solutions here in thus website? – Itachi Jul 06 '22 at 18:00
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It is an external math search engine, independent of this site, and (currently) indexes MSE and “Art of Problem Solving.” See https://math.meta.stackexchange.com/a/29267/42969 for more information. There is also a dedicated chat room – Martin R Jul 06 '22 at 18:05

For any $f\in C^1[0,1]$ show that $\sum_{k=1}^n f\big(\frac{k}{n}\big) - n \int_0^1 f(x)dx$ converges to $\frac{f(1)-f(0)}{2}$ as $n \to \infty$.

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