Let \(f(x) = \frac{8 + 5x + 12x^2}{(1-x)(2+3x)^2}\).
(a) Express \(f(x)\) in partial fractions.
(b) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\), up to and including the term in \(x^2\).
Let \(f(x) = \frac{2x(5-x)}{(3+x)(1-x)^2}\).
(i) Express \(f(x)\) in partial fractions.
(ii) Hence obtain the expansion of \(f(x)\) in ascending powers of \(x\) up to and including the term in \(x^3\).
Find the coefficient of \(x^3\) in the binomial expansion of \((3 + x)\sqrt{1 + 4x}\).
Expand \((1 - 4x)^{\frac{1}{4}}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
Expand \((3 + 2x)^{-3}\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying the coefficients.
Expand \(\frac{1}{\sqrt[3]{1 + 6x}}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
Expand \((2-x)(1+2x)^{-\frac{3}{2}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
Expand \(\frac{1}{\sqrt{1-2x}}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
Given that \(\sqrt[3]{(1 + 9x)} \approx 1 + 3x + ax^2 + bx^3\) for small values of \(x\), find the values of the coefficients \(a\) and \(b\).
Show that, for small values of \(x^2\),
\((1 - 2x^2)^{-2} - (1 + 6x^2)^{\frac{2}{3}} \approx kx^4\),
where the value of the constant \(k\) is to be determined.
Expand \((1 + 3x)^{-\frac{1}{3}}\) in ascending powers of \(x\), up to and including the term in \(x^3\), simplifying the coefficients.
Expand \(\frac{1 + 3x}{\sqrt{1 + 2x}}\) in ascending powers of \(x\) up to and including the term in \(x^2\), simplifying the coefficients.
When \((1 + ax)^{-2}\), where \(a\) is a positive constant, is expanded in ascending powers of \(x\), the coefficients of \(x\) and \(x^3\) are equal.
(i) Find the exact value of \(a\). [4]
(ii) When \(a\) has this value, obtain the expansion up to and including the term in \(x^2\), simplifying the coefficients. [3]
Expand \(\sqrt{\frac{1+2x}{1-2x}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
Expand \(\frac{1}{\sqrt{4 + 3x}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
Expand \(\sqrt{\left( \frac{1-x}{1+x} \right)}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
(i) Expand \(\frac{1}{\sqrt{1-4x}}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
(ii) Hence find the coefficient of \(x^2\) in the expansion of \(\frac{1+2x}{\sqrt{4-16x}}\).
Expand \(\frac{16}{(2+x)^2}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.
Expand \(\sqrt[3]{1 - 6x}\) in ascending powers of \(x\) up to and including the term in \(x^3\), simplifying the coefficients.
Expand \((1 + 2x)^{-3}\) in ascending powers of \(x\), up to and including the term in \(x^2\), simplifying the coefficients.