Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

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

Commented by abony1303 last updated on 15/Jun/20

pls help

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

Commented by MWSuSon last updated on 15/Jun/20

i am not sure, but i think sir used integration as the sum of infinite series where lim_(n→∞) (1/n)Σ_(k=1) ^(k=n) 𝛗((k/n))=∫_0 ^1 𝛗(x)dx

$$\boldsymbol{{i}}\:\boldsymbol{{am}}\:\boldsymbol{{not}}\:\boldsymbol{{sure}},\:\boldsymbol{{but}}\:\boldsymbol{{i}}\:\boldsymbol{{think}}\:\boldsymbol{{sir}}\:\boldsymbol{{used}} \\ $$$$\boldsymbol{{integration}}\:\boldsymbol{{as}}\:\boldsymbol{{the}}\:\boldsymbol{{sum}}\:\boldsymbol{{of}}\:\boldsymbol{{infinite}} \\ $$$$\boldsymbol{{series}}\:\boldsymbol{{where}}\:\underset{\boldsymbol{\mathrm{n}}\rightarrow\infty} {\boldsymbol{\mathrm{lim}}}\:\frac{\mathrm{1}}{\boldsymbol{{n}}}\underset{\boldsymbol{{k}}=\mathrm{1}} {\overset{\boldsymbol{{k}}=\boldsymbol{{n}}} {\sum}}\boldsymbol{\phi}\left(\frac{\boldsymbol{{k}}}{\boldsymbol{{n}}}\right)=\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\boldsymbol{\phi}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}} \\ $$

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

lim_(n→∞) (π/n)Σ_(k=1) ^n ((πk)/n)sin (((πk)/n))=π^2 ∫_0 ^1 xsin πxdx ={−πxcos πx+sin πx}_0 ^1 =𝛑 please check

$$\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}}\:\frac{\pi}{\mathrm{n}}\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\pi\mathrm{k}}{\mathrm{n}}\mathrm{sin}\:\left(\frac{\pi\mathrm{k}}{\mathrm{n}}\right)=\pi^{\mathrm{2}} \int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{xsin}\:\pi\mathrm{xdx} \\ $$$$=\left\{−\pi\mathrm{xcos}\:\pi\mathrm{x}+\mathrm{sin}\:\pi\mathrm{x}\right\}_{\mathrm{0}} ^{\mathrm{1}} =\boldsymbol{\pi}\:\:\:\:\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{check}} \\ $$

Commented by abony1303 last updated on 15/Jun/20

sorry sir, could you please explain more detailed?

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

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

let S_n =(π/n) Σ_(k=1) ^n x_k sin(x_k ) ⇒S_n =(π/n)Σ_(k=1) ^n ((kπ)/n)sin(((kπ)/n)) so S_n is a Rieman sum snd lim_(n→+∞) S_n =∫_0 ^π xsinx dx by parts ∫_0 ^π xsinx dx =[−xcosx]_0 ^π +∫_0 ^π cosx dx =π +[sinx]_0 ^π =π ⇒ lim_(n→+∞) S_n =π

$$\mathrm{let}\:\mathrm{S}_{\mathrm{n}} =\frac{\pi}{\mathrm{n}}\:\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\mathrm{x}_{\mathrm{k}} \mathrm{sin}\left(\mathrm{x}_{\mathrm{k}} \right)\:\Rightarrow\mathrm{S}_{\mathrm{n}} =\frac{\pi}{\mathrm{n}}\sum_{\mathrm{k}=\mathrm{1}} ^{\mathrm{n}} \:\frac{\mathrm{k}\pi}{\mathrm{n}}\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{n}}\right)\:\mathrm{so}\:\mathrm{S}_{\mathrm{n}} \:\mathrm{is}\:\mathrm{a}\:\mathrm{Rieman}\:\mathrm{sum} \\ $$$$\mathrm{snd}\:\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{S}_{\mathrm{n}} =\int_{\mathrm{0}} ^{\pi} \:\mathrm{xsinx}\:\mathrm{dx}\:\:\mathrm{by}\:\mathrm{parts} \\ $$$$\int_{\mathrm{0}} ^{\pi} \:\mathrm{xsinx}\:\mathrm{dx}\:=\left[−\mathrm{xcosx}\right]_{\mathrm{0}} ^{\pi} \:+\int_{\mathrm{0}} ^{\pi} \mathrm{cosx}\:\mathrm{dx}\:=\pi\:\:+\left[\mathrm{sinx}\right]_{\mathrm{0}} ^{\pi} \:=\pi\:\Rightarrow \\ $$$$\mathrm{lim}_{\mathrm{n}\rightarrow+\infty} \:\mathrm{S}_{\mathrm{n}} =\pi \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com