91Ó°ÊÓ

The given expression is ${\left(\underset{\mathrm{n}=0}{\overset{∞}{∑}}{\mathrm{r}}^{2\mathrm{n}}\mathrm{³¦´Ç²õ²Ôθ}\right)}^2+{\left(\underset{\mathrm{n}=0}{\overset{∞}{∑}}{\mathrm{r}}^{2\mathrm{n}}\mathrm{²õ¾\pm ²Ôθ}\right)}^2$.

The numbers that are presented in the form of $\mathit{a}\mathbf{+}\mathit{i}\mathit{b}$ where, a,b are real numbers and $\mathbf{\text{'}}\mathit{i}\mathbf{\text{'}}$is an imaginary number called complex numbers.

Example: 3+2i.

Consider$S=\underset{n=0}{\overset{∞}{∑}}{r}^{2n}{e}^{\left(in\mathrm{θ}\right)}$

Use Euler’s formula.

$$\begin{gathered} \begin{array}{ll}S=\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\left[\left(\cos \right(n\mathrm{θ})+i\sin \left(n\mathrm{θ}\right)\right]& \end{array} \\ \begin{array}{ll}S=\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\cos \left(n\mathrm{θ}\right)+i\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\sin \left(n\mathrm{θ}\right)& \end{array} \end{gathered}$$

Find the magnitude of the above equation.

$$\begin{gathered} \begin{array}{ll}\left|S\right|=\left|\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\cos \left(n\mathrm{θ}\right)+i\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\sin \left(n\mathrm{θ}\right)\right|& \end{array} \\ \left|S\right|=\sqrt{{\left(\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\cos \left(n\mathrm{θ}\right)\right)}^2+{\left(i\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\sin \left(n\mathrm{θ}\right)\right)}^2} \\ {\left|S\right|}^2={\left(\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\cos \left(n\mathrm{θ}\right)\right)}^2+{\left(i\underset{n=0}{\overset{∞}{∑}}{r}^{2n}\sin \left(n\mathrm{θ}\right)\right)}^2 \end{gathered}$$

Rewrite S.

$\begin{array}{l}S=\underset{n=0}{\overset{∞}{∑}}{\left[r^2{e}^{\left(i\mathrm{θ}\right)}\right]}^n\\ S=\underset{n=0}{\overset{∞}{∑}}{\left[M\right]}^n\\ S=1+{\left[r^2{e}^{\left(i\mathrm{θ}\right)}\right]}^1+{\left[r^2{e}^{\left(i\mathrm{θ}\right)}\right]}^2+{\left[r^2{e}^{\left(i\mathrm{θ}\right)}\right]}^3+···\end{array}$

Use the formula for geometric series

$$\begin{gathered} S=\frac{1}{1-M} \\ S=\frac{1}{1-\left(r^{2}exp\left(i\mathrm{θ}\right)\right)} \\ S=\frac{1}{\left[1-r^2\cos \left(\mathrm{θ}\right)\right]-i{r}^2\sin \left(\mathrm{θ}\right)} \end{gathered}$$

Find its magnitude.

$$\begin{gathered} \left|S\right|=\frac{1}{\left[1-r^2\cos \left(\mathrm{θ}\right)\right]-i{r}^2\sin \left(\mathrm{θ}\right)} \\ \left|S\right|=\frac{1}{\sqrt{{\left(1-r^2\cos \mathrm{θ}\right)}^2-{\left(r^2\sin \mathrm{θ}\right)}^2}} \\ \left|S\right|=\frac{1}{\sqrt{1-2r^2\cos {\mathrm{θ}}^2+r^4{\cos }^2\mathrm{θ}+r^4{\sin }^2\mathrm{θ}}} \\ \left|S\right|=\frac{1}{\sqrt{1-2r^2\cos \mathrm{θ}+r^4}} \\ \left|S\right|=\frac{1}{1-2r^2\cos \mathrm{θ}+r^4} \end{gathered}$$

Hence, the given criterion has been proved.