Riemann rearrangement theorem

Riemann rearrangement theorem: Let $\sum_{n=1}^\infty a_n$ be a real series that is convergent but not absolutely convergent (equivalently, it is conditionally convergent ). Then:

• For every $L\in\mathbb{R}$, there exists a bijection $\pi:\mathbb{N}\to\mathbb{N}$ such that the rearranged series $\sum_{n=1}^\infty a_{\pi(n)}$ converges to $L$.
• There also exist rearrangements that diverge to $+\infty$, diverge to $-\infty$, or fail to have a limit.

This theorem highlights the sharp difference between conditional convergence and the stability enjoyed by absolutely convergent series under rearrangement .