Riemann rearrangement theorem

Statement

Let $\sum a_n$ be a real series that converges conditionally. Let $M$ be a real number. Then there is a rearrangement $\sum a_{\sigma(n)}$ that converges to $M$. There is also a rearrangement that diverges to $+\infty$, and one that diverges to $-\infty$.

Remarks

The positive terms and the negative terms each give a divergent series, and $a_n \to 0$. Take positive terms until the partial sum exceeds $M$, then negative terms until it falls below $M$, and repeat. So a real series converges absolutely if and only if all its rearrangements converge to the same sum. For the alternating harmonic series, take two positive terms, then one negative term, and repeat. The sum becomes $\frac{3}{2} \ln 2$.

History

Augustin-Louis Cauchy is usually credited with the first remark that a new order can change a sum. He studied the alternating harmonic series in 1833. The theorem is named after Bernhard Riemann.

Explanations

L1 human

A conditionally convergent series has positive terms with an infinite total and negative terms with an infinite total. By a careful choice of order, we can steer the partial sums to any target.