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Evaluate $1+\frac i2-\frac 14-\frac i8+\frac 1{16}+\frac i{32}-\cdots$ (where $i$ is the imaginary unit).  Express your answer in the form a+bi, where a and b are real.

Mar 13, 2018

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Evaluate $1+\frac i2-\frac 14-\frac i8+\frac 1{16}+\frac i{32}-\cdots$ (where $i$ is the imaginary unit).

Express your answer in the form a+bi, where a and b are real.

$$\begin{array}{|rcll|} \hline && 1 + \dfrac{i}{2} - \dfrac{1}{4} - \dfrac{i}{8} +\dfrac{1}{16} + \dfrac{i}{32} - \dfrac{1}{64} - \dfrac{i}{128} + \cdots \\ &=& \underbrace{ 1 - \dfrac{1}{4} +\dfrac{1}{16} - \dfrac{1}{64} + \cdots}_{=\text{geom. sequence: } \\ a_1 = 1,\ r=-\dfrac{1}{4}} + \underbrace{ \dfrac{i}{2} - \dfrac{i}{8} + \dfrac{i}{32} - \dfrac{i}{128} + \cdots}_{=\text{geom. sequence: } \\ a_1 = \dfrac{i}{2},\ r=-\dfrac{1}{4}} \\ &=& 1\cdot \dfrac{1}{1-(-\dfrac{1}{4})} + \dfrac{i}{2} \cdot \dfrac{1}{1-(-\dfrac{1}{4})} \\ &=& 1\cdot \dfrac{4}{5} + \dfrac{i}{2} \cdot \dfrac{4}{5} \\ &=& \mathbf{\dfrac{4}{5} + \dfrac{2}{5} i} \\ \hline \end{array}$$

Mar 14, 2018
edited by heureka  Mar 14, 2018