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 Σan, 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 (E, q), where q ≥ 0. The (E, 1) 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 (if it converges for that value of y) corresponding to a particular formal summation as:

 _{E_y}\, \sum_{j=0}^\infty  a_j := \sum_{i=0}^\infty \frac{1}{(1+y)^{i+1}} \sum_{j=0}^i {i \choose j} y^{j+1} a_j .

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

y^{j+1}\sum_{i=j}^\infty {i \choose j} \frac{1}{(1+y)^{i+1}}=1.

This method itself cannot be improved by iterated application, as

 _{E_{y_1}} {}_{E_{y_2}}\sum = \, _{E_{\frac{y_1 y_2}{1+y_1+y_2}}} \sum.

Examples

See also

References

  • Korevaar, Jacob (2004). Tauberian Theory: A Century of Developments. Springer. ISBN 3-540-21058-X. 
  • Shawyer, Bruce and Bruce Watson (1994). Borel's Methods of Summability: Theory and Applications. Oxford UP. ISBN 0-19-853585-6. 
  • Apostol, Tom M. (1974). Mathematical Analysis Second Edition. Addison Wesley Longman. ISBN 0-201-00288-4. 
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