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Question Number 30601 by abdo imad last updated on 23/Feb/18

$$for0<r⩽1and(\theta ,x)\in R^{2}find$$ $$S=\sum _{n=0}^{\mathrm{\infty }}{r}^{n}cos\left(n\theta \right).$$

Commented byabdo imad last updated on 24/Feb/18

$$S=Re\left(\sum _{n=0}^{\mathrm{\infty }}{r}^n{e}^{in\theta }\right)but$$ $$\sum _{n=0}^{\mathrm{\infty }}{r}^n{e}^{in\theta }=\sum _{n=0}^{\mathrm{\infty }}{\left({re}^{i\theta }\right)}^n=\frac{1}{1-{re}^{i\theta }}$$ $$=\frac{1}{1-rcos\theta -irsin\theta }=\frac{1-rcos\theta +isin\theta }{{(1-rcos\theta )}^2+r^2{sin}^2\theta }\Rightarrow$$ $$S=\frac{1-rcos\theta }{{(1-rcos\theta )}^2+r^2{sin}^2\theta }=\frac{1-rcos\theta }{1-2rcos\theta +r^2}butwemust$$ $$studythecaser=1.$$

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