3
$\begingroup$

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 $\mathbb{R}$

$\endgroup$
  • $\begingroup$ 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 $\,R+ \frac{i}{2}R\,$. $\endgroup$ – dxiv Apr 3 '18 at 4:08
5
$\begingroup$

Treat real and imaginary parts separately. Note that this is rearrangement of terms does not alter the sum.

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

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.