Alternating Series: Learn It 1

  • Use the alternating series test to check if an alternating series converges
  • Understand the difference between absolute and conditional convergence

Alternating Series

Up to this point, we’ve focused on series with positive terms. Now we’ll explore alternating series—series whose terms switch between positive and negative values.

theorem: alternating series

An alternating series has terms that alternate between positive and negative values. Any alternating series can be written as:

[latex]\displaystyle\sum_{n=1}^{\infty}(-1)^{n+1}b_n = b_1 - b_2 + b_3 - b_4 + \cdots[/latex]

or

[latex]\displaystyle\sum_{n=1}^{\infty}(-1)^n b_n = -b_1 + b_2 - b_3 + b_4 - \cdots[/latex]

where [latex]b_n \geq 0[/latex] for all positive integers [latex]n[/latex].

Let’s look at two common examples:

Example 1: [latex]\displaystyle\sum_{n=1}^{\infty}\left(-\frac{1}{2}\right)^n = -\frac{1}{2} + \frac{1}{4} - \frac{1}{8} + \frac{1}{16} - \cdots[/latex]

Example 2: [latex]\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots[/latex]

The first series is a geometric series with [latex]r = -\frac{1}{2}[/latex]. Since [latex]|r| = \left|-\frac{1}{2}\right| = \frac{1}{2} < 1[/latex], this series converges.

The second series is the alternating harmonic series. While the regular harmonic series [latex]\sum_{n=1}^{\infty}\frac{1}{n}[/latex] diverges, we’ll discover that this alternating version actually converges.

The Alternating Series Test

To understand why the alternating harmonic series converges, we need to examine its sequence of partial sums [latex]\left\{{S}_{k}\right\}[/latex] (Figure 1). The key insight is to analyze the odd and even partial sums separately, showing that both subsequences converge to the same limit.

Proof

Consider the odd terms [latex]{S}_{2k+1}[/latex] for [latex]k\ge 0[/latex]. Since [latex]\frac{1}{\left(2k+1\right)}<\frac{1}{2k}[/latex],

[latex]{S}_{2k+1}={S}_{2k - 1}-\frac{1}{2k}+\frac{1}{2k+1}<{S}_{2k - 1}[/latex].

 

Therefore, [latex]\left\{{S}_{2k+1}\right\}[/latex] is a decreasing sequence. Also,

[latex]{S}_{2k+1}=\left(1-\frac{1}{2}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\cdots +\left(\frac{1}{2k - 1}-\frac{1}{2k}\right)+\frac{1}{2k+1}>0[/latex].

 

Therefore, [latex]\left\{{S}_{2k+1}\right\}[/latex] is bounded below. Since [latex]\left\{{S}_{2k+1}\right\}[/latex] is a decreasing sequence that is bounded below, by the Monotone Convergence Theorem, [latex]\left\{{S}_{2k+1}\right\}[/latex] converges. Similarly, the even terms [latex]\left\{{S}_{2k}\right\}[/latex] form an increasing sequence that is bounded above because

[latex]{S}_{2k}={S}_{2k - 2}+\frac{1}{2k - 1}-\frac{1}{2k}>{S}_{2k - 2}[/latex]

 

and

[latex]{S}_{2k}=1+\left(-\frac{1}{2}+\frac{1}{3}\right)+\cdots +\left(-\frac{1}{2k - 2}+\frac{1}{2k - 1}\right)-\frac{1}{2k}<1[/latex].

 

Therefore, by the Monotone Convergence Theorem, the sequence [latex]\left\{{S}_{2k}\right\}[/latex] also converges. Since

[latex]{S}_{2k+1}={S}_{2k}+\frac{1}{2k+1}[/latex],

 

we know that

[latex]\underset{k\to \infty }{\text{lim}}{S}_{2k+1}=\underset{k\to \infty }{\text{lim}}{S}_{2k}+\underset{k\to \infty }{\text{lim}}\frac{1}{2k+1}[/latex].

 

Letting [latex]S=\underset{k\to \infty }{\text{lim}}{S}_{2k+1}[/latex] and using the fact that [latex]\frac{1}{\left(2k+1\right)}\to 0[/latex], we conclude that [latex]\underset{k\to \infty }{\text{lim}}{S}_{2k}=S[/latex]. Since the odd terms and the even terms in the sequence of partial sums converge to the same limit [latex]S[/latex], it can be shown that the sequence of partial sums converges to [latex]S[/latex], and therefore the alternating harmonic series converges to [latex]S[/latex].

It can also be shown that [latex]S=\text{ln}2[/latex], and we can write

[latex]\displaystyle\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots =\text{ln}\left(2\right)[/latex].

 

This graph demonstrates the alternating hamanic series in the first quadrant. The highest line 1 is drawn to S1, the next line -1/2 is drawn to S2, the next line +1/3 is drawn to S3, the line -1/4 is drawn to S4, and the last line +1/5 is drawn to S5. The odd terms are decreasing and bounded below, and the even terms are increasing and bounded above. It seems to be converging to S, which is in the middle of S2, S4 and S5, S3, S1.
Figure 1. For the alternating harmonic series, the odd terms [latex]{S}_{2k+1}[/latex] in the sequence of partial sums are decreasing and bounded below. The even terms [latex]{S}_{2k}[/latex] are increasing and bounded above.

[latex]_\blacksquare[/latex]

The proof above suggests a general pattern. Any alternating series converges under specific conditions, as stated in the following theorem.

This diagram illustrates an alternating series in quadrant 1. The highest line b1 is drawn out to S1, the next line –b2 is drawn back to S2, the next line b3 is drawn out to S3, the next line –b4 is drawn back to S4, and the last line is drawn out to S5. It seems to be converging to S, which is in between S2, S4 and S5, S3, and S1. The odd terms are decreasing and bounded below. The even terms are increasing and bounded above.
Figure 2. For an alternating series [latex]{b}_{1}-{b}_{2}+{b}_{3}-\cdots [/latex] in which [latex]{b}_{1}>{b}_{2}>{b}_{3}>\cdots [/latex], the odd terms [latex]{S}_{2k+1}[/latex] in the sequence of partial sums are decreasing and bounded below. The even terms [latex]{S}_{2k}[/latex] are increasing and bounded above.

alternating series test

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]

 

converges if

  1. [latex]0\le {b}_{n+1}\le {b}_{n}[/latex] for all [latex]n\ge 1[/latex] (the terms [latex]b_n[/latex] are decreasing)
  2. [latex]\underset{n\to \infty }{\text{lim}}{b}_{n}=0[/latex] (the terms approach zero)

This is known as the alternating series test.

This theorem remains true even if the conditions only hold for [latex]n \geq N[/latex] for some integer [latex]N[/latex]. The first few terms don’t affect convergence, only the eventual behavior matters.

For each of the following alternating series, determine whether the series converges or diverges.

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