from x = 0 to the point x = 1, then the sum of the series can be defined to be the value at x = 1. This value may depend on the choice of path. One of the first examples of potentially different sums for a divergent series, using analytic continuation, was given by Callet, who observed that if then Evaluating at , one gets However, the gaps in the series are key. For for example, we actually would get , so different sums correspond to different placements of the 's. Euler summation Euler summation is essentially an explicit form of analytic continuation. If a power series converges for small complex z and can be analytically continued to the open disk with diameter from to 1 and is continuous at 1, then its value at q is called the Euler or (E,q) sum of the series Σan. Euler used it before analytic continuation was defined in general, and gave explicit formulas for the power series of the analytic continuation. The operation of Euler summation can be repeated several times, and this is essentially equivalent to taking an analytic continuation of a power series to the point z = 1. Analytic continuation of Dirichlet series This method defines the sum of a series to be the value of the analytic continuation of the Dirichlet series at s = 0, if this exists and is unique. This method is sometimes confused with zeta function regularization. If s = 0 is an isolated singularity, the sum is defined by the constant term of the Laurent series expansion. Zeta function regularization If the series (for positive values of the an) converges for large real s and can be analytically continued along the real line to s = −1, then its value at s = −1 is called the zeta regularized sum of the series a1 + a2 + ... Zeta function regularization is nonlinear. In applications, the numbers ai are sometimes the eigenvalues of a self-adjoint operator A with compact resolvent, and f(s) is then the trace of A−s. For example, if A has eigenvalues 1, 2, 3, ... then f(s) is the Riemann zeta function, ζ(s), whose value at s = −1 is −, assigning a value to the divergent series . Other values of s can also be used to assign values for the divergent sums , and in general where