Practice section 3 of 7

If $\theta$ is sufficiently small for $\theta^3$ and higher powers of $\theta$ to be neglected, find approximations for

(a) $\dfrac{\sin\theta}{\cos2\theta}$,

(b) $\dfrac{\theta\tan\theta}{1-\cos\theta}$.

If $\theta$ is sufficiently small for $\theta^3$ and higher powers of $\theta$ to be neglected, find an approximation for $\sqrt{1+\cos\theta}$.

The diagram shows a square ABCD of side AB=a cm. E is the point on BC such that angle CAE=$\theta$. Show that the area of $\triangle ACE$ is $\dfrac{\tan\theta}{1+\tan\theta}a^2$.

If $\theta$ is sufficiently small for $\theta^3$ and higher powers of $\theta$ to be neglected, show that $\operatorname{area}(\triangle ACE)\approx a^2\theta(1-\theta)$.

Using MF27, find the sum $\displaystyle\sum_{r=2}^{\infty}\frac{x^r}{r!}$, giving your answer in terms of $x$.

In the expansion of $(1+px-3x^2)^6$, the coefficient of $x^2$ is $42$. Find $p$ and the coefficient of $x^4$.

Express $f(x)=\dfrac{x^2+4x+3}{(2x+1)(x^2+1)}$ in partial fractions. Hence obtain its expansion in ascending powers of $x$ up to and including $x^3$. Find also the range of values of $x$ for which the expansion is valid.