Understanding the alternating series estimation theorem

Any series whose terms alternate between positive and negative values is called an alternating series :

\[ \sum^{\infty}_{n=1} (-1)^{n+1} b_n = b_1 - b_2 + b_3 - b_4 + b_5 - \cdots \]

Where \(b_n \gt 0\). Let's assume that \(0 \lt b_{n+1} \le b_n\) for all \(n\) and \(\lim_{n \to \infty} b_n = 0\), this means the series would converge. Let \(S\) be the value of the infinite series, let \(S_k\) be the \(k\)th partial sum, and let \(R_k = S - S_k\) (i.e. let \(R_k\) be the infinite series starting from \(n=k+1\)).

\[ S_k = \sum^{k}_{n=1} (-1)^{n+1} b_n = b_1 - b_2 + b_3 - b_4 + b_5 - \cdots ±b_k \]

Every partial sum \(S_k\) will be more any the previous even partial sums and less than any previous odd partial sums.

If \(n\) is even:

\[ S_n \le S \le S_{n+1} \]

Subtracting \(S_n\) from both sides

\[ 0 \le R_n \le S_{n+1} - S_n \]

Since \( S_{n+1} = S_n + b_{n+1} \) (keep in mind \(n\) is still even):

\[ \begin{gather} 0 \le R_n \le b_{n+1} \\ \therefore |R_n| \le b_{n+1} \text{ when } n \text{ is even}\end{gather} \]

If \(n\) is odd:

\[ S_n \ge S \ge S_{n+1} \]

Subtracting \(S_n\) from both sides

\[ 0 \ge R_n \ge S_{n+1} - S_n \]

Since \( S_{n+1} = S_n - b_{n+1} \) (keep in mind \(n\) is still odd):

\[ 0 \ge R_n \ge -b_{n+1} \]

Both \(-b_{n+1}\) and \(R_n\) are negative, so multiplying both with -1 will make them positive:

\[ \begin{gather} 0 \le -R_n \le b_{n+1} \\ |R_n| = -R_n \\ \therefore |R_n| \le b_{n+1} \text{ when } n \text{ is odd}\end{gather} \]

In either case, \(|R_n| \le b_{n+1}\) is true.

Understanding The Alternating Series Estimation Theorem