Error Estimation for Approximating Alternating Series

So far we have only looked at two methods for calculating the sums of convergent series. We have a simple technique for convergent Geometric Series, and we have a technique for Telescoping Series. We will now develop yet another technique which applies to convergent alternating series.

Theorem 1: Let $\sum_{n=1}^{\infty} a_n$ be a series satisfying all of the conditions of The Alternating Series Test, then we know that $\sum_{n=1}^{\infty} a_n = s$ for some $s \in \mathbb{R}$ (the series is convergent). The error estimation between the sum $s$ and the $n^{\mathrm{th}}$ partial sum can be evaluated by using $\mid s - s_n \mid ≤ \mid a_{n+1} \mid = \mid s_{n+1} - s_n \mid$.

The theorem above tells us that if have a series that satisfies all of the conditions of the alternating series test, and we're given some allowed error, call it $E$, then we can determine the number of terms of the series $\sum_{n=1}^{\infty} a_n$ we must evaluated in order that our partial sum $s_n$ is within the error $E$ of the actual sum $s$. Let's look at some examples.

Example 1

Determine the number of terms of the series $\sum_{n=1}^{\infty} \frac{(-1)^n}{n^2 + n}$ that are needed to be computed in order for the sum of the series to have an error less than $E = 0.001$.

The series above satisfies all three conditions of the alternating series test (verify). Using the inequality above, we need to find an $n$ such that:

(1)

\begin{align} \quad \mid s - s_n \mid ≤ \mid a_{n+1} \mid = \biggr \rvert \frac{(-1)^{n+1}}{(n+1)^2 + (n+1)} \biggr \rvert = \frac{1}{n^2 + 3n + 2} < 0.001 \end{align}

We note that this inequality holds only if the following inequality holds:

(2)

\begin{equation} n^2 + 3n + 2 > 1000 \end{equation}

We note that if $n = 31$, then $(31)^2 + 3(31) + 2 = 1060 > 1000$, and so if $n ≥ 31$ then $\mid s - s_{n} \mid < 0.001$, so the error between the partial sum $s_n$ and the actual sum $s$ is less than $0.001$.

Example 2

Determine the number of terms of the series $\sum_{n=1}^{\infty} \frac{2(-1)^n}{n}$ that are needed to be computed in order for the sum of the series to have an error less than $E = 0.01$.

Once again this series satisfies all of the conditions of the alternating series test (verify), and so we need to find an $n$ such that the following inequality holds:

(3)

\begin{align} \quad \mid s - s_n \mid ≤ \mid a_{n+1} \mid = \biggr \rvert \frac{2(-1)^{n+1}}{n+1} \biggr \rvert = \frac{2}{n+1} < 0.01 \end{align}

We note that this inequality holds if the following inequality holds:

(4)

\begin{align} \quad \frac{n+1}{2} > 100 \Leftrightarrow n > 199 \end{align}

So if $n ≥ 200$ then $\mid s - s_n \mid ≤ 0.01$ and so the error between the partial sum $s_n$ and the actual sum $s$ is less than $0.01$.