That is, it is a method for assigning a value to a series, different from the conventional method of taking limits of partial sums.
Euler summation can be generalized into a family of methods denoted (E, q), where q ≥ 0.
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: If all the formal sums actually converge, the Euler sum will equal the left hand side.
However, using Euler summation can accelerate the convergence (this is especially useful for alternating series); sometimes it can also 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