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.↩︎

$\leftarrow$ 169. Dirichlet’s Theorem Main Page 171-174. Further tests of convergence. Abel’s Theorem. Maclaurin’s integral test $\rightarrow$
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