Evaluate 1 + i/2 + 1/4 + i/8 + 1/16 + i/32 + ... (where i is the imaginary unit). Express your answer in the form a+bi, where a and b are real.
This is not a geometric sequence, but we can break it down into two.
Call the sum S. Then S = (1 + 1/4 + 1/16 + 1/64 + ...) + (i/2 + i/8 + i/32 + i/128 + ...).
Evaluating each of these separately, by the geometric series formula, the first part of the sum is 1/(1-1/4) = 4/3.
The second part of the sum is i/(2 * (1-1/4)) = 2i/3.
The answer is (4+2i)/3.