Divergent series

The Divergent series refers to a special idea in mathematics. It describes an infinite series that does not settle down to a single, finite number. When we add up the terms of such a series—one after another—the total keeps changing without ever stopping at a final value. This is different from a convergent series, where the adding-up process eventually comes close to a specific number.

One classic example of a divergent series is the harmonic series: 1 + 1/2 + 1/3 + 1/4 + 1/5 + and so on. Even though each new term gets smaller, the sum grows without bound. This was first shown by the medieval mathematician Nicole Oresme.

Sometimes, mathematicians find ways to assign a value to a divergent series even though it does not naturally settle down. They use special tools called summability methods. For instance, Cesàro summation gives the value 1/2 to the series 1 − 1 + 1 − 1 + 1 − 1, and so forth. These methods help make sense of series that otherwise would not have a clear meaning.

History

Before the 19th century, mathematicians like Leonhard Euler used divergent series, but this sometimes caused confusion. Later, Augustin-Louis Cauchy found a better way to define the sum of certain series. Divergent series became important again in the late 1800s through the work of Henri Poincaré on asymptotic series. In 1890, Ernesto Cesàro created a method called Cesàro summation to give a clear meaning to the sum of some divergent series. Since then, many mathematicians have developed different ways to understand these series.

Examples

Some examples of divergent series include:

1 - 1 + 1 - 1 + ⋯ “ = ” 1/2
1 − 2 + 3 − 4 + ⋯ “ = ” 1/4
1 − 1 + 2 − 6 + 24 − 120 + ⋯ “ = ” ≈ 0.596347…
1 − 2 + 4 − 8 + ⋯ “ = ” 1/3
1 + 2 + 4 + 8 + ⋯ “ = ” -1
1 + 1 + 1 + 1 + ⋯ “ = ” -1/2
1 + 2 + 3 + 4 + ⋯ “ = ” -1/12
1 + 4 + 9 + 16 + ⋯ “ = ” 0

These examples show how some series do not settle on a single finite value.

Theorems on methods for summing divergent series

A summability method is regular if it matches the real limit on all convergent series. Special results, called Tauberian theorems, help figure out when a series must have been convergent.

The study of divergent series looks at useful ways to handle them, like Abel summation, Cesàro summation, and Borel summation. These methods link to big ideas in math and science, such as Fourier analysis and quantum mechanics.

Properties of summation methods

Summation methods study the sequence of partial sums of a series. Even if this sequence doesn’t settle to one number, we might still find a way to assign a value to the series by averaging or other techniques.

There are special properties that summation methods can have. Regularity means the method gives the right answer for series that already have a normal sum. Linearity means the method works well with adding and scaling series. Stability means the method’s result does not change if we shift the starting point of the series. These properties help make sure the method behaves in a predictable way.

Classical summation methods

There are two main ways to find the sum of a series. These ways are called ordinary convergence and absolute convergence. These methods look at the total of the first few numbers in the series. They see what happens when you keep adding more numbers.

If these methods do not give a clear answer, the series is called divergent. This means it does not settle on one sum.

For a series to have a sum using these methods, you look at what happens when you add up more terms. Absolute convergence is a bit stricter. It looks at the total of all the terms, no matter if they are positive or negative. It checks if that total also settles down to one value. If it does, the series is absolutely convergent.

Nørlund means

The Nørlund mean is a way to find an average from a list of numbers. It uses special weights to mix the numbers together, making a new list that gets closer to a single value. This method works in a steady, straightforward way, and different Nørlund means match each other.

One important type of Nørlund mean is the Cesàro sum. These sums work in a steady, straightforward way, too. The Cesàro sum called C₀ is just regular addition, while C₁ is named Cesàro summation. If one Cesàro sum has a bigger number label than another, it is stronger.

Main article: Nørlund

Main article: Cesàro summation

Abelian means

An Abelian mean is a way to find the sum of a series that does not normally add up to a clear number. It uses a special kind of series called a generalized Dirichlet series. In simple terms, it looks at how the series behaves when a certain value gets very small, and uses that to define a new kind of sum.

One important example is Abel summation. This method changes the series into a power series by using a number z that gets closer and closer to 1. The Abel sum is found by looking at what happens to the series as z approaches 1. This method works well and agrees with another method called Cesàro summation when that method can be used.

Main article: Abel's theorem
See also: Cesàro summation
Further information: Dirichlet series, heat-kernel regularization, Mittag-Leffler star, Maclaurin series

Analytic continuation

Some ways to give meaning to a divergent series use a method called analytic continuation. This means studying a special kind of math expression that works for some numbers and then carefully extending it to other numbers.

One common example is with power series. If a series works well for small numbers and can be extended to the number 1, we can define the series' sum as the value when we reach 1. Another method uses Dirichlet series, where we look at the value when a special number s equals 0, if possible. Finally, zeta function regularization is a way to assign values to certain divergent series by studying their behavior at different points, like when s equals −1.

Main article: Zeta function regularization

Integral function means

An integral function is a special kind of math expression. There are ways to find the sum of some endless number lists. One way looks at how the function changes when numbers grow very large. Sometimes we look at how the function acts when it reaches a certain point.

There are also special methods like Borel summation. This uses different math to find the value of these sums. These methods help mathematicians understand and work with series that don’t behave in simple ways.

Main article: Borel summation

Miscellaneous methods

Some ways to give meaning to divergent series use special numbers beyond the usual real numbers. One method is called BGN hyperreal summation. Another method is Hutton's approach, which averages partial sums to find a limit.

Other techniques include Ingham summability, Lambert summability, Le Roy summation, Mittag-Leffler summation, Ramanujan summation, Riemann summability, Riesz means, Vallée-Poussin summability, and Zeldovich summability. Each method offers unique ways to interpret series that do not settle to a traditional sum.

Main article: Hölder summation Main article: Ramanujan summation Main article: Riesz mean Main article: Zeldovich regularization