1.10: Past-paper question 9

(a) Expand $\displaystyle (1 + 2x)^{-\frac{1}{3}}$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$ expressing each coefficient as a fraction in its lowest terms.

(b) Find the range of values of $\displaystyle x$ for which your expansion is valid.

(c) Obtain a series expansion of $\displaystyle f(x)$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^3$
Give your coefficients in terms of $\displaystyle k$ where appropriate.

The coefficient of $\displaystyle x^3$ in the series expansion of $\displaystyle f(x)$ is $\displaystyle -\frac{8}{3}$

(d) Find the exact value of $\displaystyle k$

(e) Hence, using algebraic integration, estimate the value of

Give your answer to 4 decimal places.