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

The first $n$ terms of Taylor's theorem, with the Taylor remainder in mean-value form, arewhere $\xi$ lies between $a$ and $x$ for $x\ne a$. Sufficient hypotheses are continuous derivatives through order $n-1$ on the interval between the endpoints and an $n$th derivative in its interior. At $x=a$ the remainder is zero. The finite Taylor polynomial ends at degree $n-1$; this identity does not require convergence of an infinite Taylor series.

Since $a_0=f(0)\ne 0$, write $1/f(x)=b_0+b_{1}x+b_2{x}^2+O(x^3)$ and multiply the two power series. Equating coefficients with the identity $f(x)/f(x)=1$ givesSolving successively yieldsThese are the constant, linear and quadratic terms; particular coefficients may vanish. For a convergent power series defining $f$ near zero, continuity and $a_0\ne 0$ ensure its reciprocal is also defined in a neighborhood of zero.

Multiply the Maclaurin series of hyperbolic cosine by the binomial series for the reciprocal square root:The quadratic terms cancel, while the quartic coefficient is $1/24-1/4+3/8=1/6$. HenceThe absence of odd powers follows from the function's even symmetry.

Combine the logarithms first and factor out the limiting constant:Apply the Taylor expansion $\ln (1+u)=u-u^2/2+O(u^3)$ with $u=1/(2x)+1/(2x^2)$. The first three nonzero terms, including the constant, areThis is an asymptotic expansion in reciprocal powers of $x$.