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Question Number 129997 by mohammad17 last updated on 21/Jan/21

$$provethat\left[(cos\left(n\right)+sin\left(n\right))/n\right]convergesequence$$

Answered by Dwaipayan Shikari last updated on 21/Jan/21

$$\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{cosn+sinn}{n}=\frac{1}{2}\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{e^{in}}{n}+\frac{e^{-in}}{n}+\frac{1}{2i}\underset{n=1}{\sum ^{\mathrm{\infty }}}\frac{e^{in}}{n}-\frac{e^{-in}}{n}$$$$=-\frac{1}{2}(log(1-2cos1+1)+log\left(\frac{1-e^i}{1-e^{-i}}\right)$$$$=\frac{1}{2}log\left(\frac{1}{4{sin}^2\frac{1}{2}}\right)+\frac{\pi }{2}-\frac{1}{2}=\frac{1}{2}(\pi -1-log\left(4{sin}^2\frac{1}{2}\right))\sim 1.11$$

Commented by mohammad17 last updated on 21/Jan/21

$$thankyousirbutiwantthisbyanalyeticfunction$$

Answered by mathmax by abdo last updated on 21/Jan/21

$$\mathrm{let}{\mathrm{u}}_{\mathrm{n}}=\frac{\cos \left(\mathrm{n}\right)+\sin \left(\mathrm{n}\right)}{\mathrm{n}}\Rightarrow {\mathrm{u}}_{\mathrm{n}}=\frac{\sqrt{2}\cos (\mathrm{n}-\frac{\pi }{4})}{\mathrm{n}}$$$$\Rightarrow \sum _{\mathrm{n}=1}^{\mathrm{\infty }}{\mathrm{u}}_{\mathrm{n}}=\sqrt{2}\sum _{\mathrm{n}=1}^{\mathrm{\infty }}\frac{\cos (\mathrm{n}-\frac{\pi }{4})}{\mathrm{n}}=\sqrt{2}\sum _{\mathrm{n}=1}^{\mathrm{\infty }}{\alpha }_{\mathrm{n}}.{\beta }_{\mathrm{n}}$$$${\alpha }_{\mathrm{n}}\mathrm{decrease}\mathrm{to}\mathrm{and}\mathrm{we}\mathrm{can}\mathrm{determine}\mathrm{m}>0/$$$$\mid \sum _{\mathrm{k}=1}^{\mathrm{n}}{\beta }_{\mathrm{k}}\mid ⩽\mathrm{m}\mathrm{so}\mathrm{the}\mathrm{convergence}\mathrm{is}\mathrm{assured}\mathrm{by}\mathrm{abel}\mathrm{dirichlet}$$$$\mathrm{theorem}..$$

Commented by mathmax by abdo last updated on 21/Jan/21

$${\alpha }_{\mathrm{n}}\mathrm{decrease}\mathrm{to}0$$

Commented by mohammad17 last updated on 21/Jan/21

$$butsirthisissequencenotseriesandmy$$$$teachersaydmeitwantthesolutionby$$$$$$$$\mathrm{\forall }\in >0\mathrm{\exists }k\in NSuchthat\mid x_n-x_0\mid <\in \mathrm{\forall }n>k$$$$$$$$canhelpme?sir$$

Commented by mathmax by abdo last updated on 22/Jan/21

$$\mathrm{your}\mathrm{teacher}\mathrm{want}\mathrm{to}\mathrm{turn}\mathrm{you}\mathrm{to}\mathrm{stone}\mathrm{era}...\mathrm{no}\mathrm{need}\mathrm{to}\mathrm{use}\xi$$$$\mathrm{look}\mathrm{we}\mathrm{have}\frac{\mathrm{cosn}+\mathrm{sinn}}{\mathrm{n}}=\frac{\sqrt{2}\cos (\mathrm{n}-\frac{\pi }{4})}{\mathrm{n}}\Rightarrow$$$$\mid \frac{\cos \left(\mathrm{n}\right)+\sin \left(\mathrm{n}\right)}{\mathrm{n}}\mid ⩽\frac{\sqrt{2}}{\mathrm{n}}\to 0\Rightarrow {\lim }_{\mathrm{n}\to +\mathrm{\infty }}\frac{\mathrm{cosn}+\mathrm{sinn}}{\mathrm{n}}=0$$

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