1.13: Binomial Series and Approximate Integration

(a) Expand

in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^6$, expressing each coefficient as an exact fraction in its lowest terms.

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

The function $\displaystyle f$ is defined by

(c) Obtain a series expansion for $\displaystyle f(x)$ in ascending powers of $\displaystyle x$ up to and including the term in $\displaystyle x^6$.

Give each coefficient in terms of $\displaystyle k$ where appropriate.

Given that the coefficient of $\displaystyle x^4$ in the series expansion of $\displaystyle f(x)$ is zero,

(d) find the value of $\displaystyle k$.

(e) Hence use algebraic integration to obtain an estimate, to 4 decimal places, of