(i) Express \(f(x)\) in partial fractions:
Assume \(\frac{10x - 2x^2}{(x+3)(x-1)^2} = \frac{A}{x+3} + \frac{B}{x-1} + \frac{C}{(x-1)^2}\).
Multiply through by \((x+3)(x-1)^2\) to clear the denominators:
\(10x - 2x^2 = A(x-1)^2 + B(x+3)(x-1) + C(x+3)\).
Expand and equate coefficients to solve for \(A\), \(B\), and \(C\):
\(A = -3, \ B = 1, \ C = 2\).
(ii) Expand in ascending powers of \(x\):
Use the expansions:
\((x+3)^{-1} = \frac{1}{3} - \frac{x}{9} + \frac{x^2}{27} + \ldots\)
\((1-x)^{-1} = 1 + x + x^2 + \ldots\)
\((1-x)^{-2} = 1 + 2x + 3x^2 + \ldots\)
Substitute and simplify:
\(\frac{-3}{x+3} = -\frac{1}{3} + \frac{x}{9} - \frac{x^2}{27} + \ldots\)
\(\frac{1}{x-1} = 1 + x + x^2 + \ldots\)
\(\frac{2}{(x-1)^2} = 2 + 4x + 6x^2 + \ldots\)
Add the series up to \(x^2\):
\(f(x) = \left(-\frac{1}{3} + \frac{x}{9} - \frac{x^2}{27}\right) + (1 + x + x^2) + (2 + 4x + 6x^2)\)
Combine terms:
\(f(x) = \frac{10}{3}x + \frac{44}{9}x^2 + \ldots\)
