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Question Number 136582 by physicstutes last updated on 23/Mar/21

$$\mathrm{Use}{\mathrm{De}}^{\prime }{\mathrm{Moivre}}^{\prime }\mathrm{s}\mathrm{theorem}\mathrm{to}\mathrm{prove}\mathrm{that}$$$$\underset{r=0}{\sum ^{\mathrm{\infty }}}\cos rx=\frac{1}{2}\mathrm{and}\mathrm{find}\mathrm{a}\mathrm{value}\left(\mathrm{or}\mathrm{expression}\right)\mathrm{for}\underset{r=0}{\sum ^{\mathrm{\infty }}}\sin rx$$$$\mathrm{assume}\mathrm{that}\mathrm{this}\mathrm{two}\mathrm{series}\mathrm{were}\mathrm{convergent}.$$

Answered by Ar Brandon last updated on 23/Mar/21

$$\underset{\mathrm{r}=0}{\sum ^{\mathrm{\infty }}}{\mathrm{e}}^{\mathrm{irx}}=\frac{1}{1-{\mathrm{e}}^{\mathrm{ix}}}=\frac{{\mathrm{e}}^{-\frac{\mathrm{x}}{2}\mathrm{i}}}{{\mathrm{e}}^{-\frac{\mathrm{x}}{2}\mathrm{i}}-{\mathrm{e}}^{\frac{\mathrm{x}}{2}\mathrm{i}}}=\frac{{\mathrm{e}}^{-\frac{\mathrm{x}}{2}\mathrm{i}}}{-2\mathrm{isin}\frac{\mathrm{x}}{2}}=\frac{{\mathrm{e}}^{-\frac{\mathrm{x}}{2}\mathrm{i}-\frac{3\pi }{2}\mathrm{i}}}{2\sin \frac{\mathrm{x}}{2}}$$$$=\frac{\cos (\frac{\mathrm{x}}{2}+\frac{3\pi }{2})-\mathrm{isin}(\frac{\mathrm{x}}{2}+\frac{3\pi }{2})}{2\sin \frac{\mathrm{x}}{2}}=\frac{\sin \frac{\mathrm{x}}{2}+\mathrm{icos}\frac{\mathrm{x}}{2}}{2\sin \frac{\mathrm{x}}{2}}$$$$\underset{\mathrm{r}=0}{\sum ^{\mathrm{\infty }}}\cos \left(\mathrm{rx}\right)=\mathrm{Re}\underset{\mathrm{r}=0}{\sum ^{\mathrm{\infty }}}{\mathrm{e}}^{\mathrm{irx}}=\frac{1}{2}$$$$\underset{\mathrm{r}=0}{\sum ^{\mathrm{\infty }}}\sin \left(\mathrm{rx}\right)=\mathrm{Im}\underset{\mathrm{r}=0}{\sum ^{\mathrm{\infty }}}{\mathrm{e}}^{\mathrm{irx}}=\frac{1}{2}\cot \left(\frac{\mathrm{x}}{2}\right)$$

Commented by physicstutes last updated on 23/Mar/21

$$\mathrm{perfect}$$

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