An **oscillating series** has a sum that wavers between one number and another. For example, the series 1 + 1 – 1 + 1, 1… wavers between 2 and 1. More formally, we would say that the limit oscillates between 2 and 1.

An oscillating series is considered to be divergent (or partially divergent), because it never reaches, or settles on a particular number (or limit).

## Oscillating Series and Partial Sums

More specifically, the partial sums of an oscillating series don’t reach a limit. Partial sums are exactly what they sound like: you **take a part of a series, and sum the numbers up**.

The reason partial sums are used is because it would be impossible to sum up an infinite series like 1 + 2 + 3…∞. You have to take a part of it (say, the first 100 numbers), otherwise you’ll be adding up to infinity (which would of course take an infinite amount of time).

## Alternate Definitions of Summation

For some oscillating series, like 1, 0, 1, 0, 1, 0, …, the limit doesn’t exist. Taking partial sums, you would never settle on a certain number. However, that **isn’t true in all cases. **

**Association of terms** in an oscillating series can lead to more than one sum, and can even lead it to converge—one reason why these series are considered to be only *partially divergent*. Take the series 1 – 1 + 1 – 1 + 1… associated in pairs:

(1 – 1) + (1 – 1) + (1 – 1) + … = 0 + 0 + 0… = 0

The series converges on zero.

Take the same series in sets of three, and the series diverges:

(1 – 1 + 1) + (-1 + 1 – 1) + (1 – 1 + 1) + … = 1 – 1 + 1 – … = 1.0.

In string theory, alternate definitions of summation are also used. This leads to the series 1, 0, 1, 0, 1, 0, …, reaching a sum of ½. For more information on that peculiar behavior (and why mathematicians in the past have called these series the “work of the devil”, see Problems with Divergent Series.

## References

Luis Manuel Braga da Costa Campos. (2010). Complex Analysis with Applications to Flows and Fields (Mathematics and Physics for Science and Technology) 1st Edition. CRC Press.