>[!abstract] >**Ramanujan summation** is a technique invented by the mathematician Srinivasa Ramanujan for assigning a value to divergent infinite series. Although the Ramanujan summation of a divergent series is not a sum in the traditional sense, it has properties that make it mathematically useful in the study of divergent infinite series, for which conventional summation is undefined (Wikipedia, 2025). >[!tip] Note >In this article, the Ramanujan summation symbol $\overset{\mathcal{R}}{=}$ is used to differentiate it from the classical sum. ## Ramanujan summation of natural numbers The infinite series whose terms are the positive integers is a divergent series (it increases without bound toward infinity). As a result, it does not have a sum. Using the Ramanujan summation, it is possible to regularize this infinite series and assign it a meaningful finite value: $\sum_{n=1}^{\infty} = 1 + 2 + 3 + \cdots \overset{\mathcal{R}}{=} -\frac{1}{12}$ ## Ramanujan summation of Grandi’s series >[!example] Additional references >- >[!related] >- **North** (upstream): — >- **West** (similar): — >- **East** (different): — >- **South** (downstream): —