Exam-Style Questions.

(a) Write down the first three terms of the binomial expansion of \( (1 - a)^{-1} \) in ascending powers of \( a \).

(b) By using the Maclaurin series for \( \dfrac{\sin x}{x} \) and the result from part (a), find the Maclaurin series for \( x\text{cosec} x \) up to and including the term in \( x^4 \).

(c) By using the Maclaurin series for \( \arctan x \) and the result from part (b) find:

$$ \lim_{{x \to 0}} \left( \frac{x \arctan 2x}{x\text{cosec}x - 1} \right) $$

Worked Solution

(a) Find the first two non-zero terms in the Maclaurin series of

(i) \( \cos(x^2) \);

(ii) \( \cos^2(x^2) \).

(b) Hence, or otherwise, find the first two non-zero terms in the Maclaurin series of

$$6x \cos(x^2) \sin(x^2)$$

Worked Solution

The function \(f\) is defined by \(f(x) = e^x \sin{x} \), where \(x \in \mathbb{R}\).

(a) Find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term.

(b) Hence, find an approximate value for \( \int_0^1 e^{x^2}\sin(x^2)dx\).

The function \(g\) is defined by \(g(x) = e^x\cos{x}\), where \(x \in \mathbb{R}\).

(c) Show that \(g(x)\) satisfies the equation \(g''(x) = 2(g'(x) — g(x)) \).

(d) Show that \(g(x)\) also satisfies the equation \(g^{(4)}(x) = 2(g'''(x) — g''(x)) \).

(e) Using these results, find the Maclaurin series for \(g(x)\) up to and including the \(x^4 \) term.

(f) Hence determine the value of \( \lim \limits_{x \to 0} \frac{e^x \cos{x} -1-x}{x^3} \)

Worked Solution