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.