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Question Number 184944 by SANOGO last updated on 14/Jan/23
please you help me  Σ_(k=0) ^n sin(k)=??
$${please}\:{you}\:{help}\:{me} \\ $$$$\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{sin}\left({k}\right)=?? \\ $$
Answered by JDamian last updated on 15/Jan/23
hint: sin(x)=((e^(xi) −e^(−xi) )/(2i))    S=Σ_(k=0) ^n sin(k)=(1/(2i))(Σ_(k=0) ^n e^(ki) −Σ_(k=0) ^n e^(−ki) )
$${hint}:\:\mathrm{sin}\left({x}\right)=\frac{{e}^{{xi}} −{e}^{−{xi}} }{\mathrm{2}{i}} \\ $$$$ \\ $$$${S}=\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\mathrm{sin}\left({k}\right)=\frac{\mathrm{1}}{\mathrm{2}{i}}\left(\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{e}^{{ki}} −\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}{e}^{−{ki}} \right) \\ $$

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