Before proceeding further in the investigation of tests of convergence and divergence, we shall prove an important general theorem concerning series of positive terms.

Dirichlet’s Theorem. The sum of a series of positive This theorem seems to have first been stated explicitly by Dirichlet in 1837. It was no doubt known to earlier writers, and in particular to Cauchy. terms is the same in whatever order the terms are taken.

This theorem asserts that if we have a convergent series of positive terms, $$u_{0} + u_{1} + u_{2} + \dots$$ say, and form any other series $v_{0} + v_{1} + v_{2} + \dots$ out of the same terms, by taking them in any new order, then the second series is convergent and has the same sum as the first. Of course no terms must be omitted: every $$u$$ must come somewhere among the $$v’$$s, and vice versa.

The proof is extremely simple. Let $$s$$ be the sum of the series of $$u’$$s. Then the sum of any number of terms, selected from the $$u’$$s, is not greater than $$s$$. But every $$v$$ is a $$u$$, and therefore the sum of any number of terms selected from the $$v’$$s is not greater than $$s$$. Hence $$\sum v_{n}$$ is convergent, and its sum $$t$$ is not greater than $$s$$. But we can show in exactly the same way that $$s \leq t$$. Thus $$s = t$$.