Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2018 / Paper 1 / 15 / c

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- Solution c

Solution

Multiply the Maclaurin series of hyperbolic cosine by the binomial series for the reciprocal square root:

cosh x = 1 + \frac{x ^{2}}{2} + \frac{x ^{4}}{24} + O (x^{6}), (1 + x^{2})^{- 1/2} = 1 - \frac{x ^{2}}{2} + \frac{3 x ^{4}}{8} + O (x^{6}) .

The quadratic terms cancel, while the quartic coefficient is

1/24 - 1/4 + 3/8 = 1/6

. Hence

\frac{cosh x}{1 + x ^{2}} = 1 + \frac{x ^{4}}{6} + O (x^{6}) .

The absence of odd powers follows from the function's even symmetry.

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