1.04d Binomial expansion validity: convergence conditions

In the binomial expansion of $$(1 - 4x)^p, \quad |x| < \frac{1}{4},$$ the coefficient of \(x^2\) is equal to the coefficient of \(x^4\) and the coefficient of \(x^3\) is positive. Find the value of \(p\). [9]

State the values of \(|x|\) for which the binomial expansion of \((3 + 2x)^{-4}\) is valid. Circle your answer. [1 mark] \(|x| < \frac{2}{3}\) \(\quad\) \(|x| < 1\) \(\quad\) \(|x| < \frac{3}{2}\) \(\quad\) \(|x| < 3\)

State the range of values of \(x\) for which the binomial expansion of $$\sqrt{1 - \frac{x}{4}}$$ is valid. Circle your answer. [1 mark] \(|x| < \frac{1}{4}\) \quad\quad \(|x| < 1\) \quad\quad \(|x| < 2\) \quad\quad \(|x| < 4\)

Write down the first three terms in the binomial expansion of \((1-4x)^{-\frac{1}{2}}\) in ascending powers of \(x\). State the range of values of \(x\) for which the expansion is valid. By writing \(x = \frac{1}{13}\) in your expansion, find an approximate value for \(\sqrt{13}\) in the form \(\frac{a}{b}\), where \(a\), \(b\) are integers. [5]

1. Use binomial expansions to show that \(\sqrt{\frac{1 + 4x}{1 - x}} \approx 1 + \frac{5}{2}x - \frac{5}{8}x^2\) [6]

A student substitutes \(x = \frac{1}{2}\) into both sides of the approximation shown in part (a) in an attempt to find an approximation to \(\sqrt{6}\)

1. Give a reason why the student should not use \(x = \frac{1}{2}\) [1]
2. Substitute \(x = \frac{1}{11}\) into $$\sqrt{\frac{1 + 4x}{1 - x}} = 1 + \frac{5}{2}x - \frac{5}{8}x^2$$ to obtain an approximation to \(\sqrt{6}\). Give your answer as a fraction in its simplest form. [3]

Let \(f(x) = \frac{1 + x^2}{\sqrt{4 - 3x}}\)

1. Obtain in ascending powers of \(x\) the first three terms in the expansion of \(\frac{1}{\sqrt{4 - 3x}}\) and state the values of \(x\) for which this expansion is valid. [5]
2. Hence obtain an approximation to \(f(x)\) in the form \(a + bx + cx^2\) where \(a\), \(b\) and \(c\) are constants. [2]
3. Use your approximation to estimate \(\int_0^{0.1} f(x) dx\). [2]