189. Abel’s and Dirichlet’s Tests of Convergence

A more general test, which includes the test of § 188 as a particular test case, is the following.

Dirichlet’s Test. If $ϕ_{n}$ satisfies the same conditions as in § 188, and $\sum a_{n}$ is any series which converges or oscillates finitely, then the series $a_{0} ϕ_{0} + a_{1} ϕ_{1} + a_{2} ϕ_{2} + \dots$ is convergent.

The reader will easily verify the identity $a_{0} ϕ_{0} + a_{1} ϕ_{1} + \dots + a_{n} ϕ_{n} = s_{0} (ϕ_{0} - ϕ_{1}) + s_{1} (ϕ_{1} - ϕ_{2}) + \dots + s_{n - 1} (ϕ_{n - 1} - ϕ_{n}) + s_{n} ϕ_{n},$ where $s_{n} = a_{0} + a_{1} + \dots + a_{n}$ . Now the series $(ϕ_{0} - ϕ_{1}) + (ϕ_{1} - ϕ_{2}) + \dots$ is convergent, since the sum to $n$ terms is $ϕ_{0} - ϕ_{n}$ and $lim ϕ_{n} = 0$ ; and all its terms are positive. Also since $\sum a_{n}$ , if not actually convergent, at any rate oscillates finitely, we can determine a constant $K$ so that $| s_{ν} | < K$ for all values of $ν$ . Hence the series $\sum s_{ν} (ϕ_{ν} - ϕ_{ν + 1})$ is absolutely convergent, and so $s_{0} (ϕ_{0} - ϕ_{1}) + s_{1} (ϕ_{1} - ϕ_{2}) + \dots + s_{n - 1} (ϕ_{n - 1} - ϕ_{n})$ tends to a limit as $n \to \infty$ . Also $ϕ_{n}$ , and therefore $s_{n} ϕ_{n}$ , tends to zero. And therefore $a_{0} ϕ_{0} + a_{1} ϕ_{1} + \dots + a_{n} ϕ_{n}$ tends to a limit, the series $\sum a_{ν} ϕ_{ν}$ is convergent.

Abel’s Test. There is another test, due to Abel, which, though of less frequent application than Dirichlet’s, is sometimes useful.

Suppose that $ϕ_{n}$ , as in Dirichlet’s Test, is a positive and decreasing function of $n$ , but that its limit as $n \to \infty$ is not necessarily zero. Thus we postulate less about $ϕ_{n}$ , but to make up for this we postulate more about $\sum a_{n}$ , viz. that it is convergent. Then we have the theorem:

if $ϕ_{n}$ is a positive and decreasing function of $n$ , and $\sum a_{n}$ is convergent, then $\sum a_{n} ϕ_{n}$ is convergent.

For $ϕ_{n}$ has a limit as $n \to \infty$ , say $l$ : and $lim (ϕ_{n} - l) = 0$ . Hence, by Dirichlet’s Test, $\sum a_{n} (ϕ_{n} - l)$ is convergent; and as $\sum a_{n}$ is convergent it follows that $\sum a_{n} ϕ_{n}$ is convergent.

This theorem may be stated as follows:

a convergent series remains convergent if we multiply its terms by any sequence of positive and decreasing factors.

Example LXXIX

1. Dirichlet’s and Abel’s Tests may also be established by means of the general principle of convergence (§ 84). Let us suppose, for example, that the conditions of Abel’s Test are satisfied. We have identically

\begin{matrix} a_{m} ϕ_{m} + a_{m + 1} ϕ_{m + 1} + \dots + a_{n} ϕ_{n} = s_{m, m} (ϕ_{m} - ϕ_{m + 1}) + s_{m, m + 1} (ϕ_{m + 1} - ϕ_{m + 2}) \\ + \dots + s_{m, n - 1} (ϕ_{n - 1} - ϕ_{n}) + s_{m, n} ϕ_{n} \dots, \Tag (1) \end{matrix}

where

s_{m, ν} = a_{m} + a_{m + 1} + \dots + a_{ν} .

The left-hand side of (1) therefore lies between $h ϕ_{m}$ and $H ϕ_{m}$ , where $h$ and $H$ are the algebraically least and greatest of $s_{m, m}$ , $s_{m, m + 1}$ , …, $s_{m, n}$ . But, given any positive number $ϵ$ , we can choose $m_{0}$ so that $| s_{m, ν} | < ϵ$ when $m \geq m_{0}$ , and so $| a_{m} ϕ_{m} + a_{m + 1} ϕ_{m + 1} + \dots + a_{n} ϕ_{n} | < ϵ ϕ_{m} \leq ϵ ϕ_{1}$ when $n > m \geq m_{0}$ . Thus the series $\sum a_{n} ϕ_{n}$ is convergent.

2. The series $\sum \cos n θ$ and $\sum \sin n θ$ oscillate finitely when $θ$ is not a multiple of $π$ . For, if we denote the sums of the first $n$ terms of the two series by $s_{n}$ and $t_{n}$ , and write $z = Cis θ$ , so that $| z | = 1$ and $z \neq 1$ , we have $| s_{n} + i t_{n} | = | \frac{1 - z^{n}}{1 - z} | \leq \frac{1 + | z^{n} |}{| 1 - z |} \leq \frac{2}{| 1 - z |};$ and so $| s_{n} |$ and $| t_{n} |$ are also not greater than $2 / | 1 - z |$ . That the series are not actually convergent follows from the fact that their $n$ th terms do not tend to zero (Ex. XXIV. 7, 8).

The sine series converges to zero if $θ$ is a multiple of $π$ . The cosine series oscillates finitely if $θ$ is an odd multiple of $π$ and diverges if $θ$ is an even multiple of $π$ .

It follows that if $θ_{n}$ is a positive function of $n$ which tends steadily to zero as $n \to \infty$ , then the series $\sum ϕ_{n} \cos n θ, \sum ϕ_{n} \sin n θ$ are convergent, except perhaps the first series when $θ$ is a multiple of $2 π$ . In this case the first series reduces to $\sum ϕ_{n}$ , which may or may not be convergent: the second series vanishes identically. If $\sum ϕ_{n}$ is convergent then both series are absolutely convergent (Ex. LXXVII. 4) for all values of $θ$ , and the whole interest of the result lies in its application to the case in which $\sum ϕ_{n}$ is divergent. And in this case the series above written are conditionally and not absolutely convergent, as will be proved in Ex. LXXIX. 6. If we put $θ = π$ in the cosine series we are led back to the result of § 188, since $\cos n π = (- 1)^{n}$ .

3. The series $\sum n^{- s} \cos n θ$ , $\sum n^{- s} \sin n θ$ are convergent if $s > 0$ , unless (in the case of the first series) $θ$ is a multiple of $2 π$ and $0 < s \leq 1$ .

4. The series of Ex. 3 are in general absolutely convergent if $s > 1$ , conditionally convergent if $0 < s \leq 1$ , and oscillatory if $s \leq 0$ (finitely if $s = 0$ and infinitely if $s < 0$ ). Mention any exceptional cases.

5. If $\sum a_{n} n^{- s}$ is convergent or oscillates finitely, then $\sum a_{n} n^{- t}$ is convergent when $t > s$ .

6. If $ϕ_{n}$ is a positive function of $n$ which tends steadily to $0$ as $n \to \infty$ , and $\sum ϕ_{n}$ is divergent, then the series $\sum ϕ_{n} \cos n θ$ , $\sum ϕ_{n} \sin n θ$ are not absolutely convergent, except the sine-series when $θ$ is a multiple of $π$ . [For suppose, , that $\sum ϕ_{n} | \cos n θ |$ is convergent. Since $\cos^{2} n θ \leq | \cos n θ |$ , it follows that $\sum ϕ_{n} \cos^{2} n θ$ or $\frac{1}{2} \sum ϕ_{n} (1 + \cos 2 n θ)$ is convergent. But this is impossible, since $\sum ϕ_{n}$ is divergent and $\sum ϕ_{n} \cos 2 n θ$ , by Dirichlet’s Test, convergent, unless $θ$ is a multiple of $π$ . And in this case it is obvious that $\sum ϕ_{n} | \cos n θ |$ is divergent. The reader should write out the corresponding argument for the sine-series, noting where it fails when $θ$ is a multiple of $π$ .]

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188. Alternating Series

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190. Series of complex terms

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