Using standard series, find the Maclaurin series expansion of $\frac{1}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}

Question

Using standard series, find the Maclaurin series expansion of $\frac{1}{\sqrt{1-{{a}^{2}}{{x}^{2}}}}$ up to and including the term in ${{x}^{4}}$, where $a$ is a positive constant. Find also the range of values of $x$, in terms of $a$, for which the expansion is valid. [5] Hence, by using a suitable value for $a$, show that ${{\cos }^{-1}}x\approx \frac{\pi }{2}-x-\frac{1}{6}{{x}^{3}}-\frac{3}{40}{{x}^{5}}$. [3] Using the result in part (b), estimate $\displaystyle\int_{0}^{\frac{1}{2}}{{{\cos }^{-1}}x\,\,\mathrm{d}x}$ to $~5$ significant figures. [2] Comparing your estimate in (c) with the actual value of $\displaystyle\int_{0}^{\frac{1}{2}}{{{\cos }^{-1}}x\,\,\mathrm{d}x}$, comment on the accuracy of your estimate and suggest how it can be further improved. [2]

Accepted Answer

(a) $\left| x ight|<\frac{1}{a}$ 	ext{ or } $-\frac{1}{a} < x < \frac{1}{a}$ since $a$ is positive. (b) ${{\cos }^{-1}}x=\frac{\pi }{2}-\frac{{{x}^{3}}}{6}-\frac{3}{40}{{x}^{5}}+...$ (c) $0.65760$ ($5$ s.f.) (d) Estimate in (c) is accurate to $4$s.f. but not to $5$ s.f. To improve estimate, we can include higher-order terms in the Maclaurin series expansion of ${{\cos }^{-1}}x$.

2023 EJC P1 Q9

2023 EJC P1 Q9

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