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# 1 − 1 + 2 − 6 + 24 − 120 + ...

In mathematics, the divergent series was first considered by Euler, who applied summability methods to assign a finite value to the series. [1]

## Asymptotic expansion

In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point.

## Borel summation

In mathematics, Borel summation is a summation method for divergent series, introduced by.

## Divergent series

In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.

## Exponential integral

In mathematics, the exponential integral Ei is a special function on the complex plane.

## Factorial

In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example, The value of 0! is 1, according to the convention for an empty product.

## Grandi's series

In mathematics, the infinite series 1 - 1 + 1 - 1 + \dotsb, also written \sum_^ (-1)^n is sometimes called Grandi's series, after Italian mathematician, philosopher, and priest Guido Grandi, who gave a memorable treatment of the series in 1703.

## Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative.

## Leonhard Euler

Leonhard Euler (Swiss Standard German:; German Standard German:; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, logician and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while also making pioneering contributions to several branches such as topology and analytic number theory.

## Power series solution of differential equations

In mathematics, the power series method is used to seek a power series solution to certain differential equations.

## 1 + 1 + 1 + 1 + ⋯

In mathematics,, also written \sum_^ n^0, \sum_^ 1^n, or simply \sum_^ 1, is a divergent series, meaning that its sequence of partial sums does not converge to a limit in the real numbers.

## 1 + 2 + 3 + 4 + ⋯

The infinite series whose terms are the natural numbers is a divergent series.

## 1 + 2 + 4 + 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two.

## 1 − 2 + 3 − 4 + ⋯

In mathematics, 1 − 2 + 3 − 4 + ··· is the infinite series whose terms are the successive positive integers, given alternating signs.

## 1 − 2 + 4 − 8 + ⋯

In mathematics, is the infinite series whose terms are the successive powers of two with alternating signs.

## References

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