Theorem 1   If

(1)

is a diverging series with positive terms, then one can always form another diverging series $$s_1\!+\!s_2\!+\!s_3\!+\cdots$$ with positive terms such that

(2)

Proof. Let $S_n = a_1\!+\!a_2\!+\cdots+\!a_n$ be the $n^\mathrm{th}$ partial sum of (1). Then we have $$a_n = S_n\!-\!S_{n-1} = (\sqrt{S_n}\!+\!\sqrt{S_{n-1}})(\sqrt{S_n}\!-\!\sqrt{S_{n-1}}).$$ We set $s_1 := \sqrt{S_1}$ and $$s_n := \frac{a_n}{\sqrt{S_n}\!+\!\sqrt{S_{n-1}}} = \sqrt{S_n}\!-\!\sqrt{S_{n-1}}$$ for $n = 2,\,3,\,4,\,\ldots$ Then the terms of the series $$\sum_{n = 1}^{\infty}s_n = \sqrt{S_1}\!+\!\sum_{n = 1}^{\infty}(\sqrt{S_{n+1}}\!-\!\sqrt{S_n})$$ apparently are positive. This series is however divergent, because the sum of its $n$ first terms is equal to $\sqrt{S_n}$ which grows without bound along with $n$ since (1) diverges. For this reason we also get the result (2).

Remark. Niels Henrik Abel has presented a simpler example on such series $s_1\!+\!s_2\!+\!s_3\!+\cdots$ : $$1\!+\!\frac{a_2}{a_1\!+\!a_2}\!+\!\frac{a_3}{a_1\!+\!a_2\!+\!a_3}\! +\!\frac{a_4}{a_1\!+\!a_2\!+\!a_3\!+\!a_4}\!+\cdots$$