Let the Maclaurin seot1q)(l2a ; ym vl3jltin7qf. ries for $\cot x$ 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 by wz g*r z11 tdqj y1j7i3yij90y $f(x)=e^{-x}\sin x-x+x^2$ .
By finding a suitable number of derivatives of f , determine the first non-zero term in its Maclaurin series.

The function f is defined sbl o46hkr)gvgj6 9lu2o) hi7 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 $\underset{x\to 0}{lim}\frac{e^x\cos x-x-1}{x^3}$.

1. By successive differentiation find the first five non-zero terms in the Maclky usvd 74a)zklc9a+)aurin series forsalu+v7 z4)kd y9c a)k $f(x)=(2-2x)\ln (1-x)+2x$ .
2. Deduce that, for $n\ge 2$, the coefficient of x^{n} in this series is $\frac{2}{n(n-1)}$.

Let $f(x)=e^x\cos x$.
1. Show that $f^{\prime \prime }(x)=2(f^{\prime }(x)-f(x))$ .
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 b3e;hdf;8f,ydrp iy ;t y $f(x)=(\arccos x{)}^2,-1\le x\le 1$.
1. Show that $f^{\prime }(0)=-\pi$

The function f satisfies the equation

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

2. By differentiating the above equation twice, show that

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

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 by aoc g15rihczd0ut s8* yhj5 )- $f(x)=e^{\arctan x}$ .
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 $\ln (2x+1)$, find the Maclaurin series for $\sin (\ln (2x+1))$ up to and including the $x^3$ term.
4. Hence, or otherwise, find $\underset{x\to 0}{lim}\frac{f(x)-1}{\sin (\ln (2x+1))}$.

Consider the functionmgk;0tqnw3-9pk yn72 nla 4i r7n,k;bgmf 8pm $f(x)=\cos (p\arccos x)$,$-1<x<1$ 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

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

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

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

4. Hence or otherwise, find $\underset{x\to 0}{lim}\frac{\cos (p\arccos (x))}{x}$ 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

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

Let $g(x)=\cos (mx),m\in \mathbb{Q}$ .
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 .