46: Applications of Partial Fractions © Christine Crisp “Teach A Level Maths” Vol. 2: A2 Core Modules

There are 2 topics where partial fractions are useful. finding some binomial expansions differentiating some algebraic fractions integrating some algebraic fractions

Method: We could write this as We would then need to expand the 2 binomial expressions, finding 4 terms for each, and finally multiply all 3 sets of brackets ! Using partial fractions means we only have to add or subtract 2 expansions. e.g. 1 Find the expansion, in ascending powers of x, up to and including the term in of the following:

Solution: Multiply by : So,

We expand the 2 fractions separately. We have: Replace n by

The 2 nd fraction is For the binomial we must have, so we take outside the brackets: so we can save time by replacing x by in that. The 1 st fraction gave so,

So, Both series must be valid so we need the most stringent condition which is

SUMMARY To find a binomial expansion for an algebraic fraction with a denominator that factorises into linear factors: Find the partial fractions. For each partial fraction, write the denominator with a negative power. Expand each part and write the values for which the series converges. Find the validity of the entire expansion by choosing the most stringent of the restrictions on x. Combine the expansions. For expressions of the form, remove from the brackets.

Exercises Express each of the following in the form given and hence obtain the expansions in ascending powers of x up to and including the term in. Give the values of x for which each expansion is valid.

1. Solutions: The partial fractions are so we need

So,

Valid for

2. The partial fractions are

By replacing x with in

For all 3 series to be valid we need