Theorem 1   If

(1)

is a converging series with positive terms, then one can always form another converging series $$g_1\!+\!g_2\!+\!g_3\!+\cdots$$ such that

(2)

Proof. Let $S$ be the sum of (1), $S_n = a_1\!+\!a_2\!+\cdots+\!a_n$ the $n^\mathrm{th}$ partial sum of (1) and $R_{n+1} = S\!-\!S_n = a_{n+1}\!+\!a_{n+2}\!+\cdots$ the corresponding remainder term. Then we have $$a_n = R_n\!-\!R_{n+1} = (\sqrt{R_n}\!+\!\sqrt{R_{n+1}})(\sqrt{R_n}\!-\!\sqrt{R_{n+1}}).$$ We set $$g_n := \frac{a_n}{\sqrt{R_n}\!+\!\sqrt{R_{n+1}}} = \sqrt{R_n}\!-\!\sqrt{R_{n+1}} \quad \forall n = 1,\,2,\,3,\,\ldots$$ Then the series $g_1\!+\!g_2\!+\!g_3\!+\cdots$ fulfils the requirements in the theorem. Its terms $g_n$ are positive. Further, it converges because its $n^\mathrm{th}$ partial sum is equal to $\sqrt{R_1}\!-\!\sqrt{R_{n+1}}$ which tends to the limit $\sqrt{R_1} = \sqrt{S}$ as $n\to\infty$ since $R_{n+1}\to 0$ ; this implies also (2).