What is the sum to infinity of the series 1 - 1/2 + 1/4 - 1/8 + .....?

Question

What is the sum to infinity of the series \(1-\frac{1}{2}+\frac{1}{4}-\frac{1}{8}\) +  .....? 

Answer 1

Given series is an infinite G.P. with a = 1, r = \(\frac{-\frac{1}{2}}{1}\) = \(-\frac{1}{2}.\)

∴ \(S_{\infty}\) = \(\frac{a}{1-r}\)

= \(\frac{1}{1-\big(-\frac{1}{2}\big)}\) = \(\frac{1}{\frac{3}{2}}\) = \(\frac{2}{3}.\)