Activity 8.3.3.
If the series \(\sum a_k\) converges, then an important result necessarily follows regarding the sequence \(\{a_n\}\text{.}\) This activity explores this result. Assume that the series \(\sum_{k=1}^{\infty} a_k\) converges and has sum equal to \(L\text{.}\)
(a)
(b)
(c)
What is the difference between the \(n\)th partial sum and the \((n-1)\)st partial sum of the series \(\sum_{k=1}^{\infty} a_k\text{?}\)
(d)
Since we are assuming that \(\sum_{k=1}^{\infty} a_k = L\text{,}\) what does that tell us about \(\lim_{n \to \infty} S_n\text{?}\) Why? What does that tell us about \(\lim_{n \to \infty} S_{n-1}\text{?}\) Why?
(e)
Combine the results of the previous two parts of this activity to determine
\begin{equation*}
\lim_{n \to \infty} a_n = \lim_{n \to \infty} (S_n - S_{n-1})\text{.}
\end{equation*}
