# Find the sum of the infinite series $1+\frac{1}{2} i+ \frac{1}{4}+ \frac{1}{8}i+\frac{1}{16}+\frac{1}{32}i+ \cdots$

Find the sum of the infinite series $1+\frac{1}{2} i+ \frac{1}{4}+ \frac{1}{8}i+\frac{1}{16}+\frac{1}{32}i+ \cdots$

I know the geometric series without the imaginary part would converge to $1$ yet I have not dealt with complex infinite series so I'm unsure how that would effect the series where I"m used to dealing only in $\mathbb{R}$

• I know the geometric series without the imaginary part would converge To use what you know about real series, calculate $\,R = 1+ \frac{1}{4}+ \frac{1}{16}+ \cdots\,$ first, then note that your series is $\,R+ \frac{i}{2}R\,$. – dxiv Apr 3 '18 at 4:08

$$S = \sum_{k=0}^{\infty}4^{-k}+i2^{-1} \sum_{k=0}^{\infty}4^{-k}\\ = \frac{4}{3}+\frac{2}{3}i$$