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}
$