Rearrangement of the Terms

If one of the two to each other complementary subseries of a convergent series is convergent the other is convergent too. A rearrangement which only shifts the two subseries relatively to each other does not change the sum of such a series.

If one of the two to each other complementary subseries of a conditionally convergent series diverges to $+\infty$ then the other diverges to $-\infty$. Provided that all the terms of one of these two subseries are of the same sign it is possible to obtain an arbitrary sum for the whole series by shifting the two subseries relatively to each other.

It is not possible to accelerate by rearrangement the divergence of a divergent series with positive monotone decreasing terms.

By rearranging the series we can slow down arbitrarily the divergence of a divergent series with positive terms which tend to $0$. More explicitly: assume

Then there exists a rearranged series $p_{\nu_1}+p_{\nu_2}+p_{\nu_3}+\cdots +p_{\nu_n}+\cdots$ such that $p_{\nu_1}+p_{\nu_2}+\cdots +p_{\nu_n} \leq Q_n$ for $n=1,2,3,\ldots$

Assume that

and $\left|a_1\right|+\left|a_2\right|+\left|a_3\right|+\cdots +\left|a_n\right|+\cdots$ is divergent. Let $s'<s<s''$. By a rearrangement that leaves all the negative terms at their places the sum $s'$ can be realized; by a rearrangement that leaves all the positive terms at their places the sum $s''$ can be realized [problem 136].