An immediate corollary from Dirichlet’s Theorem is the following theorem:

if $$u_{0} + u_{1} + u_{2} + \dots$$ and $$v_{0} + v_{1} + v_{2} + \dots$$ are two convergent series of positive terms, and $$s$$ and $$t$$ are their respective sums, then the series $u_{0} v_{0} + (u_{1} v_{0} + u_{0} v_{1}) + (u_{2} v_{0} + u_{1} v_{1} + u_{0} v_{2}) + \dots$ is convergent and has the sum $$st$$.

Arrange all the possible products of pairs $$u_{m}v_{n}$$ in the form of a doubly infinite array $\begin{array}{c|c|c|c|cc} u_{0}v_{0}& u_{1}v_{0}& u_{2}v_{0}& u_{3}v_{0}& \dots \\ {u_{0}v_{1}}& u_{1}v_{1}& u_{2}v_{1}& u_{3}v_{1}& \dots \\{u_{0}v_{2}}& {u_{1}v_{2}}& u_{2}v_{2}& u_{3}v_{2}& \dots \\{u_{0}v_{3}}& {u_{1}v_{3}}& {u_{2}v_{3}}& u_{3}v_{3}& \dots \\{\dots}&{\dots}& {\dots}& {\dots}& \dots\rlap{\;.} \end{array}$ We can rearrange these terms in the form of a simply infinite series in a variety of ways. Among these are the following.

(1) We begin with the single term $$u_{0}v_{0}$$ for which $$m + n = 0$$; then we take the two terms $$u_{1}v_{0}$$$$u_{0}v_{1}$$ for which $$m + n = 1$$; then the three terms $$u_{2}v_{0}$$$$u_{1}v_{1}$$$$u_{0}v_{2}$$ for which $$m + n = 2$$; and so on. We thus obtain the series $u_{0}v_{0} + (u_{1}v_{0} + u_{0}v_{1}) + (u_{2}v_{0} + u_{1}v_{1} + u_{0}v_{2}) + \dots$ of the theorem.

(2) We begin with the single term $$u_{0}v_{0}$$ for which both suffixes are zero; then we take the terms $$u_{1}v_{0}$$$$u_{1}v_{1}$$$$u_{0}v_{1}$$ which involve a suffix $$1$$ but no higher suffix; then the terms $$u_{2}v_{0}$$, $$u_{2}v_{1}$$, $$u_{2}v_{2}$$, $$u_{1}v_{2}$$$$u_{0}v_{2}$$ which involve a suffix $$2$$ but no higher suffix; and so on. The sums of these groups of terms are respectively equal to $\begin{gathered} u_{0}v_{0},\quad (u_{0} + u_{1})(v_{0} + v_{1}) – u_{0}v_{0},\\ (u_{0} + u_{1} + u_{2})(v_{0} + v_{1} + v_{2}) – (u_{0} + u_{1})(v_{0} + v_{1}),\ \dots\end{gathered}$ and the sum of the first $$n + 1$$ groups is $(u_{0} + u_{1} + \dots + u_{n})(v_{0} + v_{1} + \dots + v_{n}),$ and tends to $$st$$ as $$n \to \infty$$. When the sum of the series is formed in this manner the sum of the first one, two, three, … groups comprises all the terms in the first, second, third, … rectangles indicated in the diagram above.

The sum of the series formed in the second manner is $$st$$. But the first series is (when the brackets are removed) a rearrangement of the second; and therefore, by Dirichlet’s Theorem, it converges to the sum $$st$$. Thus the theorem is proved.

Example LXVIII
1. Verify that if $$r < 1$$ then $1 + r^{2} + r + r^{4} + r^{6} + r^{3} + \dots = 1 + r + r^{3} + r^{2} + r^{5} + r^{7} + \dots = 1/(1 – r).$

2.1 If either of the series $$u_{0} + u_{1} + \dots$$, $$v_{0} + v_{1} + \dots$$ is divergent, then so is the series $$u_{0}v_{0} + (u_{1}v_{0} + u_{0}v_{1}) + (u_{2}v_{0} + u_{1}v_{1} + u_{0}v_{2}) + \dots$$, except in the trivial case in which every term of one series is zero.

3. If the series $$u_{0} + u_{1} + \dots$$, $$v_{0} + v_{1} + \dots$$, $$w_{0} + w_{1} + \dots$$ converge to sums $$r$$$$s$$$$t$$, then the series $$\sum \lambda_{k}$$, where $$\lambda_{k} = \sum u_{m}v_{n}w_{p}$$, the summation being extended to all sets of values of $$m$$$$n$$$$p$$ such that $$m + n + p = k$$, converges to the sum $$rst$$.

4. If $$\sum u_{n}$$ and $$\sum v_{n}$$ converge to sums $$s$$ and $$t$$, then the series $$\sum w_{n}$$, where $$w_{n} = \sum u_{l} v_{m}$$, the summation extending to all pairs $$l$$$$m$$ for which $$lm = n$$, converges to the sum $$st$$.

1. In Exs. 2–4 the series considered are of course series of positive terms.↩︎