(i) Express \(f(x)\) as partial fractions:

Assume \(f(x) = \frac{A}{x-1} + \frac{B}{x-2} + \frac{C}{x+1}\).

Multiply through by \((x-1)(x-2)(x+1)\) to clear the denominators:

\(x^2 + 7x - 6 = A(x-2)(x+1) + B(x-1)(x+1) + C(x-1)(x-2)\).

Expand and equate coefficients to solve for \(A, B,\) and \(C\):

\(A = -1, B = 4, C = -2\).

Thus, \(f(x) = \frac{-1}{x-1} + \frac{4}{x-2} - \frac{2}{x+1}\).

(ii) Expand each term for small \(x\):

\(\frac{-1}{x-1} = -1 - x - x^2 - x^3 - \ldots\)

\(\frac{4}{x-2} = -2 - 4x - 8x^2 - 16x^3 - \ldots\)

\(\frac{-2}{x+1} = -2 + 2x - 2x^2 + 2x^3 - \ldots\)

Add these expansions:

\(f(x) = (-1 - x - x^2 - x^3) + (-2 - 4x - 8x^2 - 16x^3) + (-2 + 2x - 2x^2 + 2x^3)\)

Combine like terms:

\(f(x) = -3 + 2x - \frac{3}{2}x^2 + \frac{11}{4}x^3\).