(i) Express \(\frac{1+x}{(1-x)(2+x^2)}\) in partial fractions:

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

Multiply through by \((1-x)(2+x^2)\) to get:

\(1+x = A(2+x^2) + (Bx+C)(1-x)\).

Expand and collect terms:

\(1+x = 2A + Ax^2 + Bx - Bx^2 + C - Cx\).

Equate coefficients:

\(A = \frac{2}{3}, \ B = \frac{2}{3}, \ C = \frac{1}{3}\).

(ii) Expand \(\frac{1+x}{(1-x)(2+x^2)}\) in ascending powers of \(x\):

Use the partial fractions: \(\frac{2/3}{1-x} + \frac{(2/3)x + 1/3}{2+x^2}\).

Expand \(\frac{1}{1-x}\) as \(1 + x + x^2 + \ldots\).

Expand \(\frac{1}{2+x^2}\) as \(\frac{1}{2}(1 - \frac{1}{2}x^2 + \ldots)\).

Multiply and combine terms:

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

Combine and simplify to get:

\(\frac{1}{2} + x + \frac{3}{4}x^2\).