Stokes expressed the integral as a divergent series, which were little understood.

So using the divergent series, the sum over all harmonics is .

He concludes that the fundamental mistake is in using a divergent series to begin with:

Since then, mathematicians have explored many different summability methods for divergent series.

The term summation has a special meaning related to extrapolation in the context of divergent series.

It is a divergent series, meaning that it lacks a sum in the usual sense.

This is the basis for taking Cesàro means as a summability method in the theory of divergent series.

The harmonic series is a classic example of a divergent series whose terms limit to zero.

Using standard extensions for known divergent series, he calculated "Ramanujan summation" of those.

A simple way to sum the divergent series is by using Borel summation:

This problem involves the mathematical summation of an infinite divergent series such as Grandi's.

In mathematical analysis, Lambert summation is a summability method for a class of divergent series.

The most general method for summing a divergent series is non-constructive, and concerns Banach limits.

The asymptotic series for is usually a divergent series whose general term starts to increase after a certain value .

Euler summation is a summability method for convergent and divergent series.

Since justifications were given that used divergent series, these methods had a bad reputation from the point of view of pure mathematics.

Such divergent series can sometimes be resummed using techniques such as Borel resummation.

Summation of divergent series is also related to extrapolation methods and sequence transformations as numerical techniques.

The MRB constant is related to the following divergent series:

Krivchenkov told students that they are supposed to build up a "true" theory, not just a perturbation, that always gives a divergent series.

We suppose that ; otherwise it is obvious that is a divergent series.

However, there is a divergent series approximation that can be obtained by integrating by parts:

The first rigorous method for summing divergent series was published by Ernesto Cesàro in 1890.

One way to prove divergence is to compare the harmonic series with another divergent series:

The reciprocals of the positive integers produce a divergent series (harmonic series):