Riemann series rearrangement theorem

Statement

Consider a series:

$\sum _{k=1}^{\mathrm{\infty }}{a}_k$

(Note: the starting point of the summation does not matter for the theorem).

Suppose the series is a conditionally convergent series: it is a convergent series but not an absolutely convergent series, i.e., the series $\sum _{k=1}^{\mathrm{\infty }}\left|a_k\right|$ does not converge.

Then, the following are true:

• $\underset{k\to \mathrm{\infty }}{lim}{a}_k=0$
• The sub-series comprising only those $a_k$s that are positive diverges.
• The sub-series comprising only those $a_k$s that are negative diverges.
• Given any two elements $L\le U$ in $[-\mathrm{\infty },\mathrm{\infty }]$ (i.e., they could be real numbers, or $\pm \mathrm{\infty }$) there exists a rearrangement of the $a_k$s such that the limit inferior of the partial sums is $L$ and the limit superior of the partial sums is $U$. In particular, since we are allowed to set $L=U$, we can obtain a rearrangement that converges to any desired sum.