So far we have confined ourselves to series all of whose terms are real. We shall now consider the series $\sum u_{n} = \sum (v_{n} + iw_{n}),$ where $$v_{n}$$ and $$w_{n}$$ are real. The consideration of such series does not, of course, introduce anything really novel. The series is convergent if, and only if, the series $\sum v_{n},\quad \sum w_{n}$ are separately convergent. There is however one class of such series so important as to require special treatment. Accordingly we give the following definition, which is an obvious extension of that of § 184.

Definition. The series $$\sum u_{n}$$, where $$u_{n} = v_{n} + iw_{n}$$, is said to be absolutely convergent if the series $$\sum v_{n}$$ and $$\sum w_{n}$$ are absolutely convergent.

Theorem. The necessary and sufficient condition for the absolute convergence of $$\sum u_{n}$$ is the convergence of $$\sum |u_{n}|$$ or $$\sum \sqrt{v_{n}^{2} + w_{n}^{2}}$$.

For if $$\sum u_{n}$$ is absolutely convergent, then both of the series $$\sum |v_{n}|$$, $$\sum |w_{n}|$$ are convergent, and so $$\sum \{|v_{n}| + |w_{n}|\}$$ is convergent: but $|u_{n}| = \sqrt{v_{n}^{2} + w_{n}^{2}} \leq |v_{n}| + |w_{n}|,$ and therefore $$\sum |u_{n}|$$ is convergent. On the other hand $|v_{n}| \leq \sqrt{v_{n}^{2} + w_{n}^{2}},\quad |w_{n}| \leq \sqrt{v_{n}^{2} + w_{n}^{2}},$ so that $$\sum |v_{n}|$$ and $$\sum |w_{n}|$$ are convergent whenever $$\sum |u_{n}|$$ is convergent.

It is obvious that an absolutely convergent series is convergent, since its real and imaginary parts converge separately. And Dirichlet’s Theorem (§ 169, 185) may be extended at once to absolutely convergent complex series by applying it to the separate series $$\sum v_{n}$$ and $$\sum w_{n}$$.

The convergence of an absolutely convergent series may also be deduced directly from the general principle of convergence (cf. Ex. LXXVII. 1). We leave this as an exercise to the reader.