You can rearrange the terms of an infinite series without affecting its sum if the series is absolutely convergent.
Understanding Absolute Convergence
The key concept here is absolute convergence. An infinite series is considered absolutely convergent if the sum of the absolute values of its terms converges to a finite number. Let's break this down:
- Original Series: We have a series, like a₁ + a₂ + a₃ + ....
- Absolute Series: We take the absolute value of each term and form a new series: |a₁| + |a₂| + |a₃| + ....
- Absolute Convergence Condition: If this new series of absolute values converges to a finite number, then the original series is said to be absolutely convergent.
Why Absolute Convergence Matters for Rearrangement
When a series is absolutely convergent, it means that the terms, whether they are positive or negative, "shrink" quickly enough to ensure that when you add them up the result is a finite number. This fast shrinking prevents the sum from changing if you decide to reorder the terms, or even group them in a different way.
Formal Statement and Examples
| Series Type | Rearrangement Allowed? | Explanation |
|---|---|---|
| Absolutely Convergent | Yes | If the series of absolute values converges ( |
| Conditionally Convergent | No | If the series of absolute values does not converge (but the original series does converge), then a rearrangement can change the sum. For example, the alternating harmonic series is conditionally convergent. |
| Divergent | No | If the original series diverges, rearranging will not make it converge. |
Examples
- Absolutely Convergent Example: The series 1 + 1/2² + 1/3² + 1/4² + … is absolutely convergent. If you rearrange the order, the sum would still be the same, π²/6 .
- Conditionally Convergent Example: The series 1 - 1/2 + 1/3 - 1/4 + … is conditionally convergent. Here the sum of the absolute values is 1 + 1/2 + 1/3 + 1/4 + …, which is the harmonic series, and it diverges, even though the original series converges (to ln(2)). If you rearrange this series, the sum will change and might even be made to equal to any number.
- Divergent Example: The series 1 + 1 + 1 + 1 + … is divergent. Reordering its terms does not make it converge.
In Summary
According to the reference, you can only freely rearrange the terms of an infinite series if the series is absolutely convergent. If a series converges but is not absolutely convergent (i.e., it is conditionally convergent) then rearranging its terms can change its sum. In case the series is divergent, rearranging its terms will still not make it converge.
