Alternating Series: Learn It 2

Remainder of an Alternating Series

When working with alternating series, we often need to approximate the infinite sum using partial sums. A key question becomes: how accurate is our approximation? The good news is that for alternating series satisfying the alternating series test, we can easily bound the error.

Consider an alternating series [latex]\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}b_n[/latex] that satisfies the conditions of the alternating series test. Let [latex]S[/latex] represent the exact sum and [latex]{S_k}[/latex] be the sequence of partial sums.

From Figure 2, we can see that for any integer [latex]N \geq 1[/latex], the remainder [latex]R_N[/latex] satisfies:

[latex]|{R}_{N}|=|S-{S}_{N}|\le |{S}_{N+1}-{S}_{N}|={b}_{n+1}[/latex].

This observation leads to an important theorem.

theorem: remainders in alternating series

Consider an alternating series of the form:

[latex]\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}{b}_{n}\text{or}\displaystyle\sum _{n=1}^{\infty }{\left(-1\right)}^{n}{b}_{n}[/latex]

 

that satisfies the hypotheses of the alternating series test. Let [latex]S[/latex] denote the sum of the series and [latex]{S}_{N}[/latex] denote the [latex]N\text{th}[/latex] partial sum. For any integer [latex]N\ge 1[/latex], the remainder [latex]{R}_{N}=S-{S}_{N}[/latex] satisfies:

[latex]|{R}_{N}|\le {b}_{N+1}[/latex].
What This Means: If the alternating series test conditions are met, the error in approximating the infinite series by the [latex]N[/latex]th partial sum is at most the size of the very next term [latex]b_{N+1}[/latex]. This makes error estimation remarkably straightforward!

Consider the alternating series

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{{n}^{2}}[/latex].

 

Use the remainder estimate to determine a bound on the error [latex]{R}_{10}[/latex] if we approximate the sum of the series by the partial sum [latex]{S}_{10}[/latex].