(i) Express \(f(x)\) in partial fractions:
Assume \(\frac{2x(5-x)}{(3+x)(1-x)^2} = \frac{A}{3+x} + \frac{B}{1-x} + \frac{C}{(1-x)^2}\).
Multiply through by \((3+x)(1-x)^2\) to clear the denominators:
\(2x(5-x) = A(1-x)^2 + B(3+x)(1-x) + C(3+x)\).
Expand and equate coefficients to find \(A = -3\), \(B = -1\), \(C = 2\).
(ii) Expand \(f(x)\) in ascending powers of \(x\):
Use the expansions:
\((3+x)^{-1} = \frac{1}{3} - \frac{1}{9}x + \frac{1}{27}x^2 + \cdots\)
\((1-x)^{-1} = 1 + x + x^2 + \cdots\)
\((1-x)^{-2} = 1 + 2x + 3x^2 + \cdots\)
Substitute into the partial fractions:
\(\frac{-3}{3+x} = -1 + \frac{1}{3}x - \frac{1}{9}x^2 + \cdots\)
\(\frac{-1}{1-x} = -1 - x - x^2 + \cdots\)
\(\frac{2}{(1-x)^2} = 2 + 4x + 6x^2 + \cdots\)
Add the expansions and simplify:
\(f(x) = \left(-1 + \frac{1}{3}x - \frac{1}{9}x^2 + \cdots \right) + \left(-1 - x - x^2 + \cdots \right) + \left(2 + 4x + 6x^2 + \cdots \right)\)
Combine like terms:
\(f(x) = \frac{10}{3}x + \frac{44}{9}x^2 + \frac{190}{27}x^3\).
