Question and Answers Forum

All Questions      Topic List

Integration Questions

Previous in All Question      Next in All Question      

Previous in Integration      Next in Integration      

Question Number 121498 by Lordose last updated on 08/Nov/20

Evaluate Σ_(n=1) ^∞ ((sin(n))/n)

$$ \\ $$$$\mathrm{Evaluate} \\ $$$$\underset{\mathrm{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{sin}\left(\mathrm{n}\right)}{\mathrm{n}} \\ $$

Answered by Bird last updated on 09/Nov/20

let find S(x) =Σ_(n=1) ^(∞ ) ((sin(nx))/n)dx S(x)=Im(Σ_(n=1) ^∞ (e^(inx) /n)) and Σ_(n=1) ^(∞ ) (e^(inx) /n) =Σ_(n=1) ^(∞ ) (((e^(ix) )^n )/n) =−ln(1−e^(ix) ) =−ln(1−cosx−isinx ) =−ln(2sin^2 ((x/2))−2isin((x/2))cos((x/2))) =−ln(−2isin((x/2))e^((ix)/2) ) =−ln(−2)−ln(i)−ln(sin((x/2)))−((ix)/2) =−ln(2)−iπ−((iπ)/2)−((ix)/2)−ln(sin((x/2))) =−ln2−ln(sin((x/2)))−((ix)/2)−((3iπ)/2) ⇒Σ ((sin(nx))/n) =−((x+3π)/2) x=1 ⇒ Σ ((sin(n))/n) =−((3π+1)/2)

$${let}\:{find}\:{S}\left({x}\right)\:=\sum_{{n}=\mathrm{1}} ^{\infty\:} \:\frac{{sin}\left({nx}\right)}{{n}}{dx} \\ $$$${S}\left({x}\right)={Im}\left(\sum_{{n}=\mathrm{1}} ^{\infty} \:\frac{{e}^{{inx}} }{{n}}\right)\:{and} \\ $$$$\sum_{{n}=\mathrm{1}} ^{\infty\:} \:\frac{{e}^{{inx}} }{{n}}\:=\sum_{{n}=\mathrm{1}} ^{\infty\:} \:\frac{\left({e}^{{ix}} \right)^{{n}} }{{n}} \\ $$$$=−{ln}\left(\mathrm{1}−{e}^{{ix}} \right) \\ $$$$=−{ln}\left(\mathrm{1}−{cosx}−{isinx}\:\right) \\ $$$$=−{ln}\left(\mathrm{2}{sin}^{\mathrm{2}} \left(\frac{{x}}{\mathrm{2}}\right)−\mathrm{2}{isin}\left(\frac{{x}}{\mathrm{2}}\right){cos}\left(\frac{{x}}{\mathrm{2}}\right)\right) \\ $$$$=−{ln}\left(−\mathrm{2}{isin}\left(\frac{{x}}{\mathrm{2}}\right){e}^{\frac{{ix}}{\mathrm{2}}} \right) \\ $$$$=−{ln}\left(−\mathrm{2}\right)−{ln}\left({i}\right)−{ln}\left({sin}\left(\frac{{x}}{\mathrm{2}}\right)\right)−\frac{{ix}}{\mathrm{2}} \\ $$$$=−{ln}\left(\mathrm{2}\right)−{i}\pi−\frac{{i}\pi}{\mathrm{2}}−\frac{{ix}}{\mathrm{2}}−{ln}\left({sin}\left(\frac{{x}}{\mathrm{2}}\right)\right) \\ $$$$=−{ln}\mathrm{2}−{ln}\left({sin}\left(\frac{{x}}{\mathrm{2}}\right)\right)−\frac{{ix}}{\mathrm{2}}−\frac{\mathrm{3}{i}\pi}{\mathrm{2}} \\ $$$$\Rightarrow\Sigma\:\frac{{sin}\left({nx}\right)}{{n}}\:=−\frac{{x}+\mathrm{3}\pi}{\mathrm{2}} \\ $$$${x}=\mathrm{1}\:\Rightarrow\:\Sigma\:\:\frac{{sin}\left({n}\right)}{{n}}\:=−\frac{\mathrm{3}\pi+\mathrm{1}}{\mathrm{2}} \\ $$$$ \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com