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Question Number 98687 by abony1303 last updated on 15/Jun/20

Commented by abony1303 last updated on 15/Jun/20

$$\mathrm{pls}\mathrm{help}$$

Commented by MWSuSon last updated on 15/Jun/20

$$\mathit{i}\mathit{a}\mathit{m}\mathit{n}\mathit{o}\mathit{t}\mathit{s}\mathit{u}\mathit{r}\mathit{e},\mathit{b}\mathit{u}\mathit{t}\mathit{i}\mathit{t}\mathit{h}\mathit{i}\mathit{n}\mathit{k}\mathit{s}\mathit{i}\mathit{r}\mathit{u}\mathit{s}\mathit{e}\mathit{d}$$$$\mathit{i}\mathit{n}\mathit{t}\mathit{e}\mathit{g}\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{a}\mathit{s}\mathit{t}\mathit{h}\mathit{e}\mathit{s}\mathit{u}\mathit{m}\mathit{o}\mathit{f}\mathit{i}\mathit{n}\mathit{f}\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{e}$$$$\mathit{s}\mathit{e}\mathit{r}\mathit{i}\mathit{e}\mathit{s}\mathit{w}\mathit{h}\mathit{e}\mathit{r}\mathit{e}\underset{\mathbf{n}\to \mathrm{\infty }}{\mathbf{lim}}\frac{1}{\mathit{n}}\underset{\mathit{k}=1}{\sum ^{\mathit{k}=\mathit{n}}}\mathit{\varphi }\left(\frac{\mathit{k}}{\mathit{n}}\right)=\underset{0}{\stackrel{1}{\int }}\mathit{\varphi }\left(\mathit{x}\right)\mathit{d}\mathit{x}$$

Commented by PRITHWISH SEN 2 last updated on 15/Jun/20

$$\underset{\mathrm{n}\to \mathrm{\infty }}{\lim }\frac{\pi }{\mathrm{n}}\underset{\mathrm{k}=1}{\sum ^{\mathrm{n}}}\frac{\pi \mathrm{k}}{\mathrm{n}}\sin \left(\frac{\pi \mathrm{k}}{\mathrm{n}}\right)={\pi }^2{\int }_0^1\mathrm{xsin}\pi \mathrm{xdx}$$$$={\{-\pi \mathrm{xcos}\pi \mathrm{x}+\sin \pi \mathrm{x}\}}_0^1=\mathit{\pi }\mathbf{please}\mathbf{check}$$

Commented by abony1303 last updated on 15/Jun/20

$$\mathrm{sorry}\mathrm{sir},\mathrm{could}\mathrm{you}\mathrm{please}\mathrm{explain}\mathrm{more}$$$$\mathrm{detailed}?$$

Answered by mathmax by abdo last updated on 15/Jun/20

$$\mathrm{let}{\mathrm{S}}_{\mathrm{n}}=\frac{\pi }{\mathrm{n}}\sum _{\mathrm{k}=1}^{\mathrm{n}}{\mathrm{x}}_{\mathrm{k}}\sin \left({\mathrm{x}}_{\mathrm{k}}\right)\Rightarrow {\mathrm{S}}_{\mathrm{n}}=\frac{\pi }{\mathrm{n}}\sum _{\mathrm{k}=1}^{\mathrm{n}}\frac{\mathrm{k}\pi }{\mathrm{n}}\sin \left(\frac{\mathrm{k}\pi }{\mathrm{n}}\right)\mathrm{so}{\mathrm{S}}_{\mathrm{n}}\mathrm{is}\mathrm{a}\mathrm{Rieman}\mathrm{sum}$$$$\mathrm{snd}{\lim }_{\mathrm{n}\to +\mathrm{\infty }}{\mathrm{S}}_{\mathrm{n}}={\int }_0^{\pi }\mathrm{xsinx}\mathrm{dx}\mathrm{by}\mathrm{parts}$$$${\int }_0^{\pi }\mathrm{xsinx}\mathrm{dx}={[-\mathrm{xcosx}]}_0^{\pi }+{\int }_0^{\pi }\mathrm{cosx}\mathrm{dx}=\pi +{\left[\mathrm{sinx}\right]}_0^{\pi }=\pi \Rightarrow$$$${\lim }_{\mathrm{n}\to +\mathrm{\infty }}{\mathrm{S}}_{\mathrm{n}}=\pi$$$$$$

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