Activity 8.2.4.
The formulas we have derived for an infinite geometric series and its partial sum have assumed we begin indexing the sums at \(n=0\text{.}\) If instead we have a sum that does not begin at \(n=0\text{,}\) we can factor out common terms and use the established formulas. This process is illustrated in the examples in this activity.
(a)
Consider the sum
\begin{equation*}
\sum_{k=1}^{\infty} (2)\left(\frac{1}{3}\right)^k = (2)\left(\frac{1}{3}\right) + (2)\left(\frac{1}{3}\right)^2 + (2)\left(\frac{1}{3}\right)^3 + \cdots\text{.}
\end{equation*}
Remove the common factor of \((2)\left(\frac{1}{3}\right)\) from each term and hence find the sum of the series.
(b)
Next let \(a\) and \(r\) be real numbers with \(-1\lt r\lt 1\text{.}\) Consider the sum
\begin{equation*}
\sum_{k=3}^{\infty} ar^k = ar^3+ar^4+ar^5 + \cdots\text{.}
\end{equation*}
Remove the common factor of \(ar^3\) from each term and find the sum of the series.
(c)
Finally, we consider the most general case. Let \(a\) and \(r\) be real numbers with \(-1\lt r\lt 1\text{,}\) let \(n\) be a positive integer, and consider the sum
\begin{equation*}
\sum_{k=n}^{\infty} ar^k = ar^n+ar^{n+1}+ar^{n+2} + \cdots\text{.}
\end{equation*}
Remove the common factor of \(ar^n\) from each term to find the sum of the series.
