Euler summation

Euler summation is a summability method for convergent and divergent series

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 limit....

. Given a series Σa_n, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series.

Euler summation can be generalized into a family of methods denoted (E, q), where q ≥ 0. The (E, 0) sum is the usual (convergent) sum, while (E, 1) is the ordinary Euler sum. All of these methods are strictly weaker than Borel summation; for q > 0 they are incomparable with Abel summation.

Definition

Euler summation is particularly used to accelerate the convergence

Series acceleration

In mathematics, series acceleration is one of a collection of sequence transformations for improving the rate of convergence of a series. Techniques for series acceleration are often applied in numerical analysis, where they are used to improve the speed of numerical integration...

 of alternating series and allows evaluating divergent sums.

To justify the approach notice that for interchanged sum, Euler's summation reduces to the initial series, because

This method itself cannot be improved by iterated application, as

Examples

• We have , if is a polynomial of degree
  Degree of a polynomial
  The degree of a polynomial represents the highest degree of a polynominal's terms , should the polynomial be expressed in canonical form . The degree of an individual term is the sum of the exponents acting on the term's variables...
  
   k. Note that in this case Euler summation reduces an infinite series to a finite sum.

• The particular choice provides an explicit representation of the Bernoulli numbers, since . Indeed, applying Euler summation to the zeta function yields , which is polynomial for a positive integer; cf. Riemann zeta function.

• . With an appropriate choice of this series converges to .

Definition

If has a continuous derivative on , then we have
where is the greatest integer which is less than or equal to .
Proof: Applying the Riemann-Stieltjes integral

Riemann-Stieltjes integral

In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.-Definition:...

 integration by parts, we have

Since the greatest-integer function has unit jumps at the integers we can write

If we combine this fact with the previous equation, we have

Now we rearrange this equation to get the following

When and are integers, this becomes

.

NOTE: means the sum from to

Examples

• if

• 

See also

• Borel summation
• Cesàro summation
  Cesàro summation
  In mathematical analysis, Cesàro summation is an alternative means of assigning a sum to an infinite series. If the series converges in the usual sense to a sum A, then the series is also Cesàro summable and has Cesàro sum A...
  
  
• Lambert summation
  Lambert summation
  In mathematical analysis, Lambert summation is a summability method for a class of divergent series.-Definition:A series \sum a_n is Lambert summable to A, written \sum a_n = A , if...
  
  
• Abelian and tauberian theorems
  Abelian and tauberian theorems
  In mathematics, abelian and tauberian theorems are theorems giving conditions for two methods of summing divergent series to give the same result, named after Niels Henrik Abel and Alfred Tauber...
  
  
• Abel's summation formula
• Van Wijngaarden transformation