Question and Answers Forum

All Questions      Topic List

Limits Questions

Previous in All Question      Next in All Question      

Previous in Limits      Next in Limits      

Question Number 128846 by Ar Brandon last updated on 10/Jan/21

u_n =Σ_(k=1) ^n sin(((kπ)/n))sin((k/n^2 ))  Find lim_(n→∞) u_n

$$\mathrm{u}_{\mathrm{n}} =\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{n}}\right)\mathrm{sin}\left(\frac{\mathrm{k}}{\mathrm{n}^{\mathrm{2}} }\right) \\ $$$$\mathrm{Find}\:\underset{\mathrm{n}\rightarrow\infty} {\mathrm{lim}u}_{\mathrm{n}} \\ $$

Commented by Dwaipayan Shikari last updated on 10/Jan/21

lim_(n→∞) sin((k/n^2 ))=(k/n^2 )  u_n =lim_(n→∞) (1/n)Σ_(k=1) ^n (k/n)sin((kπ)/n)  =∫_0 ^1 xsinxπ dx = (1/π^2 )∫_0 ^π u sinu =(1/π^2 )[−u cosu]_0 ^π +(1/π^2 )∫_0 ^π cosu du  =(1/π)

$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}{sin}\left(\frac{{k}}{{n}^{\mathrm{2}} }\right)=\frac{{k}}{{n}^{\mathrm{2}} } \\ $$$${u}_{{n}} =\underset{{n}\rightarrow\infty} {\mathrm{lim}}\frac{\mathrm{1}}{{n}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\frac{{k}}{{n}}{sin}\frac{{k}\pi}{{n}} \\ $$$$=\int_{\mathrm{0}} ^{\mathrm{1}} {xsinx}\pi\:{dx}\:=\:\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}} ^{\pi} {u}\:{sinu}\:=\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\left[−{u}\:{cosu}\right]_{\mathrm{0}} ^{\pi} +\frac{\mathrm{1}}{\pi^{\mathrm{2}} }\int_{\mathrm{0}} ^{\pi} {cosu}\:{du} \\ $$$$=\frac{\mathrm{1}}{\pi} \\ $$

Commented by Dwaipayan Shikari last updated on 10/Jan/21

(1/π)=((2(√2))/(9801))Σ_(k=0) ^∞ (((4k)!(1103+26390k))/((k!)^4 396^(4k) ))

$$\frac{\mathrm{1}}{\pi}=\frac{\mathrm{2}\sqrt{\mathrm{2}}}{\mathrm{9801}}\underset{{k}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\left(\mathrm{4}{k}\right)!\left(\mathrm{1103}+\mathrm{26390}{k}\right)}{\left({k}!\right)^{\mathrm{4}} \mathrm{396}^{\mathrm{4}{k}} } \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com