The value of sum_{n=0}^{infty}left(frac{2 i}{ | vedclass

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(A) The given series is an infinite geometric series with the first term $a = 1$ and common ratio $r = \frac{2 i}{3}$.Since $|r| = |\frac{2 i}{3}| = \

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$\frac{9+6 i}{13}$

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(A) The given series is an infinite geometric series with the first term $a = 1$ and common ratio $r = \frac{2 i}{3}$.Since $|r| = |\frac{2 i}{3}| = \frac{2}{3} $S = \frac{1}{1-\frac{2 i}{3}} = \frac{1}{\frac{3-2 i}{3}} = \frac{3}{3-2 i}$.To simplify,multiply the numerator and denominator by the conjugate $(3+2 i)$:$S = \frac{3(3+2 i)}{(3-2 i)(3+2 i)} = \frac{9+6 i}{3^2 - (2 i)^2} = \frac{9+6 i}{9 - (-4)} = \frac{9+6 i}{13}$.

The value of $\sum_{n=0}^{\infty}\left(\frac{2 i}{3}\right)^n$ is

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