Mathematics

Problem

The geometric series

G=\sum_{n=0}^{\infty}\left(\frac{-1}{2}\right)^n

can be written in closed form. A different infinite series

H

is produced by listing the first two positive terms of

G

, then the first negative term, then the next two positive terms, then the next negative term, and so on. Specifically,

H=1+\frac14-\frac12+\frac1{16}+\frac1{32}-\frac1{8}+\frac1{64}+\frac1{128}-\frac1{32}+\cdots

(a) Determine

\displaystyle\sum_{n=0}^{\infty}\left(\frac{-1}{2}\right)^n

. (b) Decide whether

H

converges. If it converges, find its sum; if it diverges, state why. (c) Justify your answer in part (b) by analyzing the relationship between

G

and

H

. (c) Justify your answer in part (b) by analyzing the relationship between

G

and

H

. Explicitly reference the structure of the terms in

H

. No further numeric calculations are required beyond what you performed in part (a).