In this question, you may use expansions from the List of Formulae and Results (MF$27$). It is giv

Question

In this question, you may use expansions from the List of Formulae and Results (MF$27$). It is given that $a > 0$. Find, in terms of $a$, the series expansion of $\frac{a}{a-x}-1$, 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.[4] 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 $x^2$.[2] Use the expansion in part (ii) to approximate $\displaystyle \displaystyle\int_0^{\frac{a}{2}} \mathrm{e}^{\frac{a}{a-x}-1} \, \mathrm{d}x$. Explain why this approximation is an under-estimation.[4]

Accepted Answer

(i)$\frac{x}{a}+\frac{x^2}{a^2}+...$, -a< x < a (ii)$1+\frac{x}{a}+\frac{3}{2}\frac{x^2}{a^2}+...$ (iii)$\frac{11}{16}a$

2025 NJC P1 Q8

2025 NJC P1 Q8