Rearrangement of a series

A rearrangement of a series $\sum_{n=1}^\infty a_n$ is a new series of the form $\sum_{n=1}^\infty a_{\sigma(n)}$, where $\sigma$ is a bijection of the positive integers.

For absolutely convergent series , every rearrangement converges to the same value (see the rearrangement theorem for absolutely convergent series ), while for conditionally convergent series the behavior can change drastically (see the Riemann rearrangement theorem ).

Examples:

• Any rearrangement of $\sum_{n=0}^\infty \frac{1}{2^n}$ converges to the same sum.
• Rearrangements of $\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n}$ can converge to different sums, or even diverge, depending on the permutation.