Euler summation


In the mathematics of convergent and divergent series, Euler summation is a summability method. That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums. Given a series ∑an, if its Euler transform converges to a sum, then that sum is called the Euler sum of the original series. As well as being used to define values for divergent series, Euler summation can be used to speed the convergence of series.
Euler summation can be generalized into a family of methods denoted, where q ≥ 0. The sum 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

For some value y we may define the Euler sum corresponding to a particular formal summation as:
If the formal sum actually converges, an Euler sum will equal it. But Euler summation is particularly used to accelerate the convergence of alternating series and sometimes it can give a useful meaning to 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

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