1.14: Past-paper question 9

(a) Expand $\displaystyle (1-8x^2)^{\frac{1}{2}}$ in ascending powers of $\displaystyle x$, up to and including the term in $\displaystyle x^6$ giving each coefficient as an integer.

Given that the fourth and fifth terms, in ascending powers of $\displaystyle x$, in the series expansion of $\displaystyle g(x)$ are $\displaystyle 20x^3$ and $\displaystyle 48x^4$ respectively,

(b) find the value of $\displaystyle a$ and the value of $\displaystyle b$

Using the first five terms, in ascending powers of $\displaystyle x$, in the series expansion of $\displaystyle g(x)$

(c) obtain an estimate, to 4 significant figures, of $\displaystyle \int_0^{0.2} g(x) \, \mathrm{d}x$