College Algebra, Chapter 9, 9.3, Section 9.3, Problem 48

Determine the sum $\displaystyle 1 - \frac{1}{2} + \frac{1}{4} - \frac{1}{8} + ... - \frac{1}{512}$.

Here the geometric sequence has $a = 1$ and $\displaystyle r = \frac{-1}{2}$, using the formula


$\begin{equation} \begin{aligned} a_n =& ar^{n -1} \\ \\ 512 =& 1(2)^{n -1} \\ \\ \ln 512 =& \ln 2^{n -1} \\ \\ n - 1 =& \frac{\ln 512}{\ln 2} \\ \\ n =& \frac{\ln 512}{\ln 2} + 1 \\ \\ n =& 10 \end{aligned} \end{equation}$



From the formula of geometric partial sum


$\begin{equation} \begin{aligned} S_n =& a \frac{1 - r^n}{1 - r} \\ \\ S_{10} =& 1 \left( \frac{\displaystyle 1 - \left( \frac{-1}{2} \right)^{10} }{\displaystyle 1 - \left( \frac{-1}{2} \right) } \right) \\ \\ S_{10} =& \frac{1025}{1536} \end{aligned} \end{equation}$