Alternating Series

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Alternating Series.

Special among infinite series which contain both positive and negative terms are those whose terms alternate in sign. More precisely, we define the series [math]\sum_{i=m}^\infty a_i[/math] to be \textbf{alternating} if [math]a_{i}a_{i+1} \lt 0[/math] for every integer [math]i \geq m[/math]. It follows from this definition that an alternating series is one which can be written in one of the two forms

[[math]] \sum_{i=1}^\infty (-1)^{i}b_{i}\;\;\; \mbox{or} \;\;\; \sum_{i=m}^\infty (-1)^{i+1}b_{i}, [[/math]]

where [math]b_i \gt 0[/math] for every integer [math]i \geq m[/math]. An example is the alternating harmonic series

[[math]] \sum_{i=1}^\infty (-1)^{i+1} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} - \cdots . [[/math]]

An alternating series converges under surprisingly weak conditions. The next theorem gives two simple hypotheses whose conjunction is sufficient to imply convergence.

Theorem

The alternuting series [math]\sum_{i=m}^\infty a_i[/math] concerges if:

\item[i]] [math]|a_{n + 1}| \leq |a_n|, \;\;\; \mathrm{for every integer} n \geq m, \;\mathrm{and}[/math] \item[(ii)] [math]\lim_{n \rightarrow \infty} a_n = 0 \;\mathrm{(or, equivalently,} \; \lim_{n \rightarrow \infty} |a_n| = 0).[/math]

Show Proof

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As an application of Theorem (4.1) consider the alternating harmonic series

[[math]] \sum_{ i=1}^\infty (-1)^{i + 1} \frac{1}{i} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots . [[/math]]

The hypotheses of the theorem are obviously satisfied:

\item[(i)] [math]\frac{1}{n + 1} \leq \frac{1}{n}, \;\;\; \mbox{for every integer [/math]n \geq 1[math], and}[/math] \item[(ii)] [math]\lim_{n \rightarrow \infty} (-1)^{n+1} \frac{1}{n} = \lim_{n \rightarrow \infty} \frac{1}{n} = 0.[/math]

Hence it follows that the alternating harmonic series is convergent. It is interesting to compare this series with the ordinary harmonic series [math] \sum_{i=1}^\infty \frac{1}{i} = 1 + \frac{1}{2} + \frac{1}{3} +\frac{1}{4} + \cdots [/math], which we have shown to be divergent. We see that the alternating harmonic series is a convergent infinite series [math]\sum_{i=m}^{\infty} a_i[/math] for which the corresponding series of absolute values [math]\sum_{i=m}^{\infty} |a_i|[/math] fail diverges. For practical purposes, the value of a convergent infinite series [math]\sum_{i=m}^{\infty} a_i[/math] is usually approximated by a partial [math]\sum_{i=m}^{\infty} a_i[/math]. The error in the approximation, denoted by [math]E_n[/math], is the absolute value of the difference between the true value of the series and the approximating partial sum; i.e.,

[[math]] E_n = | \sum_{i=m}^{\infty} a_i - \sum_{i=m}^{n} a_i | . [[/math]]

ln general, it is a difficult problem to know how large [math]n[/math] must be chosen to cosure that the error [math]E_n[/math] be less than a given size. However, for those alternating series which satisfy the hypotheses of Theorem (4.1), the problem is an easy one.

Theorem

If the ulternating series [math]\sum_{i=m}^{\infty} a_i[/math] satisfies hypotheses (i) and (ii) of Theorem (4.1), then the error [math]E_n[/math] is less than or equal to the absolute value of the first omitted term. That is,

[[math]] E_n \leq |a_{n+1}|, \;\;\; \mbox{for every integer} \; n \geq m. [[/math]]

Show Proof

We shall use the same notation as in the proof of (4.1). Thus we assume that [math] m = 0[/math] and that [math]a_i = (-1)^{i} b_i[/math] where [math]b_i \gt 0[/math] for every integer [math]i \geq 0[/math]. The value of the series is the number [math]L[/math], and the error [math]E_n[/math] is therefore given by

[[math]] E_n = | \sum_{i=0}^{\infty} a_i - \sum_{i=0}^{n} a_i | = |L - s_n| . [[/math]]

Since [math]|a_{n+1}| = b_{n+1}[/math], the proof is completed by showing that

[[math]] |L - s_n| \leq b_{n+1}, \; \mbox{for every integer}\; n \geq 0. [[/math]]

Geometrically, [math]|L - s_n|[/math] is the distance between the points [math]L[/math] and [math]s_n[/math] and it can be seen immediately from Figure 5 that the preceding inequality is true. To arrive at the conclusion formally, we recall that [math]\{s_{2n-1} \}[/math] is an increasing sequence converging to [math]L[/math], and that [math]\{s_{2n} \}[/math] is a decreasing sequence converging to [math]L[/math]. Thus if [math]n[/math] is odd, then [math]n + 1[/math] is even and

[[math]] s_n \leq L \leq s_{n+1}. [[/math]]

On the other hand, if [math]n[/math] is even, then [math]n + 1[/math] is odd and

[[math]] s_{n +1} \leq L \leq s_n. [[/math]]

In either case, we have [math]|L - s_n| \leq |s_{n+1} - s_n|[/math]. Hence, for every integer [math]n \geq 0[/math],

[[math]] E_n = |L - s_n| \leq |s_{n+1} - s_n| = |a_{n+1}|, [[/math]]

and the proof is complete.

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In Table 1 we have computed some partial sums which approximate the value of the alternating harmonic series. Each entry in the second column is an approximation, and the corresponding entry in the third column is the upper bound on the error provided by Theorem (4.2).

TABLE 1

\multicolumn{3}{c}{Alternating harmonic series: [math]\sum_{i=1}^\infty (-1)^{i+1} \frac{1}{i}[/math] . }
\multicolumn{3}{c}{Partial sums: [math]s_n = 1 - \frac{1}{2} + \frac{1}{3} - \cdots + (-1)^{n+1} \frac{1}{n} [/math]. }
[math]n[/math]	[math]s_n[/math] = approximation	[math]|a_{n+1}| = \frac{1}{n + 1}[/math] = upper bound for error
1	1	[math]\frac{1}{2}[/math]
2	[math]\frac{1}{2}[/math]	[math]\frac{1}{3}[/math]
3	[math]\frac{5}{6}[/math]	[math]\frac{1}{4}[/math]
4	[math]\frac{7}{12}[/math]	[math]\frac{1}{5}[/math]
10	0.6460	0.0910
100	0.6882	0.0099
1000	0.6926	0.0010
10,000	0.6931	0.0001

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\end{exercise}

General references

Doyle, Peter G. (2008). "Crowell and Slesnick's Calculus with Analytic Geometry" (PDF). Retrieved Oct 29, 2024.