Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2006 / Paper 1 / 9 / b / iii / Solution

The removable value at zero is one. Expand the denominator as

e^{x} - 1 = x (1 + x /2 + x^{2} /6 + \dots)

and determine the reciprocal coefficients by multiplication. The first four nonzero terms are

\frac{x}{e ^{x} - 1} = 1 - \frac{x}{2} + \frac{x ^{2}}{12} - \frac{x ^{4}}{720} + O (x^{6}) .

This is the exponential generating function of the Bernoulli numbers, with convention

B_{1} = - 1/2

. The coefficients of

x^{3}

and

x^{5}

are zero.

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