Question

Express (5x+2)/(2x-1)(x+1) into partial fractions and hence expand the expression as a series in ascending power of x giving the first 4 terms

Answer 1

(5x+2)/(2x-1)(x+1)
= 3/(2x-1) + 1/(x+1)

Now, think of geometric series where the infinite sum is
S = a/(1-r)

3/(2x-1) = -3/(1-2x) = -3(1+(2x)+(2x)^2 + (2x)^3)
= -3(1+2x+4x^2+8x^3)
= -3 - 6x - 12x^2 - 24x^3

Similarly for 1/(x+1) so we end up with


-2 - 7x - 11x^2 - 25x^3 - ...