1.12: Past-paper question 7

(a) Expand $\displaystyle \left(1 + \frac{x}{3}\right)^{-3}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$

Where appropriate express each coefficient as an exact fraction in its lowest terms.

(b) Write down the range of values of $\displaystyle x$ for which your expression is valid.

(c) Express $\displaystyle (3 + x)^{-3}$ in the form $\displaystyle P(1 + Qx)^{-3}$ where $\displaystyle P$ and $\displaystyle Q$ are rational numbers whose values should be stated.

(d) Obtain a series expansion for $\displaystyle f(x)$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^2$

(e) Hence, using algebraic integration, obtain an estimate of $\displaystyle \int_0^{0.2} f(x) \, \mathrm{d}x$

Give your answer to 5 significant figures.