You can rearrange the terms of a series without changing its sum if the series is absolutely convergent. If the series is conditionally convergent, rearranging the terms will change the sum (or may even cause the series to diverge). This is known as the Riemann Rearrangement Theorem.
Absolutely vs. Conditionally Convergent Series
To understand when you can rearrange a series, you need to know the difference between absolute and conditional convergence:
- Absolutely Convergent Series: A series ∑an is absolutely convergent if the series of the absolute values of its terms, ∑|an|, converges.
- Conditionally Convergent Series: A series ∑an is conditionally convergent if it converges, but the series of the absolute values of its terms, ∑|an|, diverges.
The Riemann Rearrangement Theorem
The Riemann Rearrangement Theorem, named after Bernhard Riemann, states:
If a series ∑an of real numbers is conditionally convergent, then for any real number L, there exists a rearrangement of the series such that the rearranged series converges to L. Furthermore, there exists a rearrangement that diverges to +∞, one that diverges to -∞, and one that oscillates.
In essence, this means that if a series is only conditionally convergent, rearranging the order of its terms can drastically alter its behavior.
Why Absolute Convergence Matters
The reason absolute convergence allows rearrangement is that it guarantees that the "positive part" and the "negative part" of the series both converge individually. This allows you to move terms around without affecting the overall limit. If the series is only conditionally convergent, the "positive part" and "negative part" both diverge, and the way you interleave them dictates the limit.
Example
Consider the alternating harmonic series:
1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + ...
This series is conditionally convergent; it converges to ln(2). However, the harmonic series (1 + 1/2 + 1/3 + 1/4 + ...) diverges. The Riemann Rearrangement Theorem says we can rearrange this series to converge to any real number, or even diverge.
In Summary
- Absolutely Convergent Series: You can rearrange the terms without changing the sum.
- Conditionally Convergent Series: Rearranging the terms will generally change the sum or make the series diverge.
