In our work in

Section 8.2, we learned how to find a degree

\(n\) polynomial approximation centered at a value

\(a\) for a given function

\(f\) with at least

\(n\) derivatives. By working with several different functions and

\(n\)-values, we’ve seen that increasing the degree of the polynomial improves the approximation, and also often helps us to see a pattern in the coefficients of the Taylor polynomials.

For example, the degree \(5\) Taylor approximation of \(f(x) = \ln(1+x)\) at \(a = 0\) is

\begin{equation*}
T_5(x) = x - \frac{1}{2}x^2 + \frac{1}{3}x^3 - \frac{1}{4}x^4 + \frac{1}{5}x^5,
\end{equation*}

and we see a pattern in the coefficients that allows us to easily generate \(T_6(x)\text{,}\) \(T_{10}(x)\text{,}\) or indeed \(T_n(x)\) for any \(n\text{.}\) Note that if we want to use \(T_{10}(x)\) to estimate \(\ln(\frac{3}{2}) = \ln(1 + \frac{1}{2})\text{,}\) we need to compute the sum of \(10\) terms given by

\begin{equation*}
\ln\left( \frac{3}{2} \right) \approx T_{10}\left( \frac{1}{2} \right) = \left( \frac{1}{2} \right) - \frac{1}{2} \left( \frac{1}{2} \right)^2 + \frac{1}{3} \left( \frac{1}{2} \right)^3 - \frac{1}{4} \left( \frac{1}{2} \right)^4 + \cdots - \frac{1}{10} \left( \frac{1}{2} \right)^{10}.
\end{equation*}

This computation suggests at least two questions: is there an easy way (without a computer) to determine the exact value of the \(10\)-term sum

\begin{equation*}
\left( \frac{1}{2} \right) - \frac{1}{2} \left( \frac{1}{2} \right)^2 + \frac{1}{3} \left( \frac{1}{2} \right)^3 - \frac{1}{4} \left( \frac{1}{2} \right)^4 + \cdots - \frac{1}{10} \left( \frac{1}{2} \right)^{10}\text{,}
\end{equation*}

and is there a way we can make sense of continuing the sum indefinitely,

\begin{equation*}
\left( \frac{1}{2} \right) - \frac{1}{2} \left( \frac{1}{2} \right)^2 + \frac{1}{3} \left( \frac{1}{2} \right)^3 - \frac{1}{4} \left( \frac{1}{2} \right)^4 + \cdots - \frac{1}{10} \left( \frac{1}{2} \right)^{10} + \cdots?
\end{equation*}

In this section, we investigate a special collection of similar sums that are called geometric.