Chapter 10.1, Problem 8E

To determine

To compute the limit of partial sums to determine whether the series converges or diverges.

Answer to Problem 8E

Series converges to $1$

Explanation of Solution

Given:

The given series is $\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\mathrm{......}+\frac{1}{2^k}+\mathrm{......}$

Calculation:

The given series is an example of sum to infinite terms of GP.

The sum can be calculated as follows-

  $a=\frac{1}{2},r=\frac{1}{2}$

So,

  $\begin{array}{l}{S}_{\infty }=\frac{a}{1-r}=\frac{\frac{1}{2}}{1-\frac{1}{2}}\\ =\frac{\frac{1}{2}}{\frac{1}{2}}=1\end{array}$

Therefore the series converges to $1$