Wikipedia:
"In mathematics, the Riemann series theorem (also called the Riemann rearrangement theorem), named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an arbitrary real number, or diverges."
Question of mine is:
If $\sum a_k$ is some conditionally convergent series and $c \in (- \infty, + \infty)$ then is there at least a countably infinite number of permutations $\sigma_m(c);m=1,2,…$ such that $\sum a_{\sigma_m(c)}=c$ for every $c \in (- \infty, + \infty)$?
Best Answer
Yes. Let the sum of the first $n$ terms of your series be $s_n$. Then the remainder of the series is conditionally convergent, so it can be rearranged to achieve a sum of $c-s_n$. Choose one, and then move on. For each $n$ this yields at least $n!$ permutations that give the desired sum, no more than $n$ of which can have been duplicated at a prior step.