Let the Maclaurin series frtvw h/h 5f22uor $\cot x$ be

$\cot x=\frac{a_{1}}{x}+a_{2} x+a_{3} x^{3}+\cdots$

where $a_{1}$,$ a_{2} $ and $a_{3} $ are non zero constants.
1. Find the series for $\csc ^{2} x$ , in terms of $a_{1}$,$ a_{2} $ and $a_{3} $, up to and including the $x^{2}$ term
a. by differentiating the above series for $\cot x$ ;
b. by using the relationship $\csc ^{2} x=1+\cot ^{2} x$ .
2. Hence, by comparing your two series, determine the values of $a_{1}$,$ a_{2} $ and $a_{3} $ .

The function f is defined bj2 ljo/f7dek 5y $f(x)=e^{-x} \sin x-x+x^{2}$ .
By finding a suitable number of derivatives of f , determine the first non-zero term in its Maclaurin series.

The function f is definede.ulcm)x+i kon( k6wi ;n+ks6/9 jrmg by $ f(x)=e^{x} \cos x, x \in \mathbb{R} $.
1. By finding a suitable number of derivatives of f , determine the Maclaurin series for f(x) as far as the term $x^{4} $.
2. Hence, or otherwise, determine the exact value of $\lim _{x \rightarrow 0} \frac{e^{x} \cos x-x-1}{x^{3}} $.

1. By successive diffc3vq . ; 6up+3w,b scjz;iebl/5 rrbhmerentiation find the first five non-zero terms in the Maclaurip,mcc w ev6r;.r5u3blqjh bs3;/ +zbin series for $ f(x)=(2-2 x) \ln (1-x)+2 x$ .
2. Deduce that, for $n \geq 2 $, the coefficient of x^{n} in this series is $\frac{2}{n(n-1)} $.

Let $f(x)=e^{x} \cos x $.
1. Show that $ f^{\prime \prime}(x)=2\left(f^{\prime}(x)-f(x)\right)$ .
2. By further differentiation of the result in part (a), find the Maclaurin expansion of f(x) , as far as the term in $x^{5} $.

The function f is defined p;l-c/ rns pt84 cq6wfby $ f(x)=(\arccos x)^{2},-1 \leq x \leq 1 $.
1. Show that $ f^{\prime}(0)=-\pi $

The function f satisfies the equation

$\left(1-x^{2}\right) f^{\prime \prime}(x)-x f^{\prime}(x)=2$ .

2. By differentiating the above equation twice, show that

$\left(1-x^{2}\right) f^{(4)}-5 x f^{\prime \prime \prime}(x)=4 f^{\prime \prime}(x)$

where f^{(n)}(x) denotes the n th derivative of f(x) .
3. Hence show the Maclaurin series for f(x) up to and including the term in $x^{4}$ is $\frac{\pi^{2}}{4}-\pi x+x^{2}-\frac{\pi}{6} x^{3}+\frac{x^{4}}{3}$ .
4. Use this series approximation for f(x) with $x=\frac{1}{2}$ to find an approximate value for $25 \pi-\frac{20}{3} \pi^{2}$ .

The function f is defined ;lno-dsf6*h(eqp sjul(lx ax:9; )o f by $f(x)=e^{\arctan x}$ .
1. Find the first two derivatives of f(x) and hence find the Maclaurin series for f(x) up to and including the $x^{2} $ term.
2. Show that the coefficient of $x^{3}$ in the Maclaurin series for f(x) is $-\frac{1}{6}$ .
3. Using the Maclaurin series for $\sin x $ and $\ln (2 x+1) $, find the Maclaurin series for $\sin (\ln (2 x+1)) $ up to and including the $ x^{3}$ term.
4. Hence, or otherwise, find $\lim _{x \rightarrow 0} \frac{f(x)-1}{\sin (\ln (2 x+1))} $.

Consider the functionc .zoj) xrjc8uw 0t*o( $ f(x)=\cos (p \arccos x)$,$-1\lt x\lt 1 $ and$ p \in \mathbb{R} $.
1. Show that $ f^{\prime}(0)=p \sin \left(\frac{p \pi}{2}\right)$.

The function f and its derivative satisfy

$\left(1-x^{2}\right) f^{(n+2)}(x)=(2 n+1) x f^{(n+1)}(x)+\left(n^{2}-p^{2}\right) f^{(n)}(x), \quad n \in \mathbb{N}$

where f^{(n)}(x) denotes the n th derivative of $f(x)$ and $ f^{(0)}(x) $ is f(x) .
2. Show that $f^{(n+2)}(0)=\left(n^{2}-p^{2}\right) f^{(n)}(0)$ .
3. For $ p \in \mathbb{R} \backslash\{ \pm 1, \pm 2, \pm 3\}$ , show that the Maclaurin series for f(x) , up to and including the x^{4} term, is

$\begin{aligned}
\cos \left(\frac{p \pi}{2}\right)+p & \sin \left(\frac{p \pi}{2}\right) x-\frac{p^{2} \cos \left(\frac{p \pi}{2}\right)}{2} x^{2} \\
+ & \frac{\left(1-p^{2}\right) p \sin \left(\frac{p \pi}{2}\right)}{6} x^{3}-\frac{\left(4-p^{2}\right) p^{2} \cos \left(\frac{p \pi}{2}\right)}{24} x^{4}
\end{aligned}$

4. Hence or otherwise, find $\lim _{x \rightarrow 0} \frac{\cos (p \arccos (x))}{x} $ where p is an odd integer.
5. If p is an integer, prove the Maclaurin series for f(x) is a polynomial of .

Let $f(x)=\frac{1}{\sqrt{1-x}}$, x<1 .
1. Show that $f^{\prime \prime}(x)=\frac{3}{4}(1-x)^{-5 / 2} $.
2. Use mathematical induction to prove that

$f^{(n)}(x)=\left(\frac{1}{4}\right)^{n} \frac{(2 n)!}{n!}(1-x)^{-1 / 2-n} \quad n \in \mathbb{Z}, \quad n \geq 2 $.

Let $g(x)=\cos (m x), m \in \mathbb{Q}$ .
Consider the function h defined by $ h(x)=f(x) \times g(x) $ for x<1 .
The x^{2} term in the Maclaurin series for h(x) has a coefficient of $-\frac{3}{4}$ .
3. Find the possible values of m .