ABSTRACT
This series diverges for all finite values of x, and therefore appears to be meaningless. Why did the procedure succeed in the first case but not in the second? The answer
is not hard to find. The expansion of cost converges for all values of I ; indeed it converges uniformly throughout any bounded t interval. Application of a standard theorem concerning integration of an infinite series over an infinite interval' confirms that the step from (1.02) to (1.03) is completely justified when x > 1. In the second example, however, the expansion of (1 +t)- ' diverges when t 2 1. The failure of the representation (1.05) may be regarded as the penalty for integrating a series over an interval in which it is not uniformly convergent. 1.2 If our approach to mathematical analysis were one of unyielding purity, then we might be content to leave these examples at this stage. Suppose, however, we adopt a heuristic approach and try to sum the series (1.05) numerically for a particular value of x, say x = 10. The first four terms are given by
exactly, and the sum of the series up to this point is 0.0914. Somewhat surprisingly this is very close to the correct value G(10) = 0.09156 ... .!
