One can recall that the sum of an infinite series in the Cauchy sense was the limit of partial sums.

We consider a very general and powerful method which relies on an alternative in taking the limit.

Suppose s_n is a partial sum of the series a_0 + a_1 + a_2 + ... which is under consideration and that S(t) = (p_0)(s_0) + (p_1)(s_1).t + (p_2)(s_2).t^2 + .... is entire for non-negative coefficients p_n and P(t) = (p_0) + (p_1).t + (p_2).t^2 + ...

If lim_{t->infinity} S(t) / P(t) = s

then we say,

a_0 + a_1 + a_2 + ... = s , (P)

(while we have used P to refer to this summation method, it ought not be considered with Abel summation for which P is sometimes used for Poisson who also employed it in his investigations into Fourier series. Hardy in Divergent Series calls this the J method pp79-80)

We put p_n = 1/ (n!) and thus P(t) = e^t to define the method of Borel summation. That is if,

lim_{t->infinity} e^(-t) S(t)

= lim_{t->infinity} e^(-t) [(s_0) + (s_1).t + (s_2).(t^2)/2! + ... + (s_n).(t^n)/n! + ...] = s

then,

a_0 + a_1 + a_2 + ... = s , (B)

Since we have,

n! = integral_{t = 0}^{infinity} (t^n).e^(-t) dt

, by employing a property of the Gamma function we have a similar summation,

a_0 + a_1 + a_2 + ...

= sum_{n = 0}^{infinity} (a_n).[integral_{t = 0}^{infinity} (t^n).e^(-t) dt] / n!

= integral_{t = 0}^{infinity} [sum_{n = 0}^{infinity} (a_n).(t^n)/n!] e^(-t) dt = s

(the interchange of summation and integration is allowed given the convergence of n!)

Should this last integral converge and the series within has a non-zero radius of convergence we say,

a_0 + a_1 + a_2 + ... = s , (B')

Should e^(-t) [sum_{n = 0}^{infinity} (a_n).(t^n)/n!] -> 0 as t -> infinity the methods B and B' may be easily shown to be equivalent.

See Hardy, Divergent Series pp182-183

The Borel methods certainly allow us to sum more types of series than we had been able to thus far, for instance, it is easily shown that the B sum of 1 + z + z^2 + z^3 + ... = 1 / (1 - z) throughout the half plane Re(z) < 1

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