Maclaurin Series: binomial expansion, composite exponential and series integration — NJC 2025 H2 Math Prelim P1
What this question tests
Question
In this question, you may use expansions from the List of Formulae and Results (MF27).
It is given that \(a>0\).
(i) Find, in terms of \(a\), the series expansion of \(\dfrac{a}{a-x}-1\), in ascending powers of \(x\), up to and including the term in \(x^2\). State, in terms of \(a\), the range of \(x\) for which the expansion is valid.
(ii) Hence, find the Maclaurin expansion of \(\mathrm{e}^{\frac{a}{a-x}-1}\) in ascending powers of \(x\), up to and including the term in \(x^2\).
(iii) Use the expansion in part (ii) to approximate \(\displaystyle\int_0^{a/2}\mathrm{e}^{\frac{a}{a-x}-1}\,\mathrm{d}x\). Explain why this approximation is an under-estimation.
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(i) Binomial expansion and validity
Rewrite the fraction as \((1-\tfrac{x}{a})^{-1}\) and apply the binomial series, then subtract \(1\).
\[ \begin{aligned} \frac{a}{a-x} - 1 &= a(a-x)^{-1} - 1 \\ &= \left(1-\frac{x}{a}\right)^{-1} - 1 \\ &= \left[1+\left(-\frac{x}{a}\right)+\left(-\frac{x}{a}\right)^{2}+\cdots\right] - 1 \\ &= \frac{x}{a}+\frac{x^2}{a^2}+\cdots \end{aligned} \]Validity: \(\left|-\dfrac{x}{a}\right|<1 \Rightarrow |x| \(\dfrac{a}{a-x}-1 \approx \dfrac{x}{a}+\dfrac{x^2}{a^2}+\cdots\), valid for \(-a (ii) Composite exponential expansion Substitute the series from (i) into \(\mathrm{e}^{u}=1+u+\tfrac{1}{2}u^2+\cdots\), keeping terms up to \(x^2\). \(\mathrm{e}^{\frac{a}{a-x}-1} \approx 1 + \dfrac{x}{a} + \dfrac{3x^2}{2a^2} + \cdots\) (iii) Approximating the integral Integrate the polynomial approximation term by term from \(0\) to \(a/2\). Neglected terms for \(0