184–185. Absolutely convergent series

184. Absolutely Convergent Series.

Let us then consider a series $\sum u_{n}$ in which any term may be either positive or negative. Let \[|u_{n}| = \alpha_{n},\] so that $\alpha_{n} = u_{n}$ if $u_{n}$ is positive and $\alpha_{n} = -u_{n}$ if $u_{n}$ is negative. Further, let $v_{n} = u_{n}$ or $v_{n} = 0$, according as $u_{n}$ is positive or negative, and $w_{n} = -u_{n}$ or $w_{n} = 0$, according as $u_{n}$ is negative or positive; or, what is the same thing, let $v_{n}$ or $w_{n}$ be equal to $\alpha_{n}$ according as $u_{n}$ is positive or negative, the other being in either case equal to zero. Then it is evident that $v_{n}$ and $w_{n}$ are always positive, and that \[u_{n} = v_{n} – w_{n},\quad \alpha_{n} = v_{n} + w_{n}.\]

If, for example, our series is $1 – (1/2)^{2} + (1/3)^{2} – \dots$, then $u_{n} = (-1)^{n-1}/n^{2}$ and $\alpha_{n} = 1/n^{2}$, while $v_{n} = 1/n^{2}$ or $v_{n} = 0$ according as $n$ is odd or even and $w_{n} = 1/n^{2}$ or $w_{n} = 0$ according as $n$ is even or odd.

We can now distinguish two cases.

A. Suppose that the series $\sum \alpha_{n}$ is convergent. This is the case, for instance, in the example above, where $\sum \alpha_{n}$ is \[1 + (1/2)^{2} + (1/3)^{2} + \dots.\] Then both $\sum v_{n}$ and $\sum w_{n}$ are convergent: for (Ex. XXX. 18) any series selected from the terms of a convergent series of positive terms is convergent. And hence, by theorem (6) of § 77, $\sum u_{n}$ or $\sum(v_{n} – w_{n})$ is convergent and equal to $\sum v_{n} – \sum w_{n}$.

We are thus led to formulate the following definition.

When $\sum \alpha_{n}$ or $\sum |u_{n}|$ is convergent, the series $\sum u_{n}$ is said to be absolutely convergent.

And what we have proved above amounts to this:

if $\sum u_{n}$ is absolutely convergent then it is convergent; so are the series formed by its positive and negative terms taken separately; and the sum of the series is equal to the sum of the positive terms plus the sum of the negative terms.

The reader should carefully guard himself against supposing that the statement ‘an absolutely convergent series is convergent’ is a mere tautology. When we say that $\sum u_{n}$ is ‘absolutely convergent’ we do not assert directly that $\sum u_{n}$ is convergent: we assert the convergence of another series $\sum |u_{n}|$, and it is by no means evident a priori that this precludes oscillation on the part of $\sum u_{n}$.

Example LXXVII

1. Employ the ‘general principle of convergence’ (§ 84) to prove the theorem that an absolutely convergent series is convergent. [Since $\sum |u_{n}|$ is convergent, we can, when any positive number $\epsilon$ is assigned, choose $n_{0}$ so that \[|u_{n_{1}+1}| + |u_{n_{1}+2}| + \dots + |u_{n_{2}}| < \epsilon\] when $n_{2} > n_{1} \geq n_{0}$. A fortiori \[|u_{n_{1}+1} + u_{n_{1}+2} + \dots + u_{n_{2}}| < \epsilon,\] and therefore $\sum u_{n}$ is convergent.]

2. If $\sum a_{n}$ is a convergent series of positive terms, and $|b_{n}|\leq Ka_{n}$, then $\sum b_{n}$ is absolutely convergent.

3. If $\sum a_{n}$ is a convergent series of positive terms, then the series $\sum a_{n}x^{n}$ is absolutely convergent when $-1 \leq x \leq 1$.

4. If $\sum a_{n}$ is a convergent series of positive terms, then the series $\sum a_{n} \cos n\theta$, $\sum a_{n}\sin n\theta$ are absolutely convergent for all values of $\theta$. [Examples are afforded by the series $\sum r^{n}\cos n\theta$, $\sum r^{n}\sin n\theta$ of § 88.]

5. Any series selected from the terms of an absolutely convergent series is absolutely convergent. [For the series of the moduli of its terms is a selection from the series of the moduli of the terms of the original series.]

6. Prove that if $\sum |u_{n}|$ is convergent then \[|\sum u_{n}| \leq \sum |u_{n}|,\] and that the only case to which the sign of equality can apply is that in which every term has the same sign.

185. Extension of Dirichlet’s Theorem to absolutely convergent series.

Dirichlet’s Theorem (§ 169) shows that the terms of a series of positive terms may be rearranged in any way without affecting its sum. It is now easy to see that any absolutely convergent series has the same property. For let $\sum u_{n}$ be so rearranged as to become $\sum u’_{n}$, and let $\alpha’_{n}$, $v’_{n}$, $w’_{n}$ be formed from $u’_{n}$ as $\alpha_{n}$, $v_{n}$, $w_{n}$ were formed from $u_{n}$. Then $\sum \alpha’_{n}$ is convergent, as it is a rearrangement of $\sum \alpha_{n}$, and so are $\sum v’_{n}$, $\sum w’_{n}$, which are rearrangements of $\sum v_{n}$, $\sum w_{n}$. Also, by Dirichlet’s Theorem, $\sum v’_{n} = \sum v_{n}$ and $\sum w’_{n} = \sum w_{n}$ and so \[\sum u’_{n} = \sum v’_{n} – \sum w’_{n} = \sum v_{n} – \sum w_{n} = \sum u_{n}.\]

$\leftarrow$ 183. Series of positive and negative terms

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186–187. Conditionally convergent series $\rightarrow$