Hence or otherwise, determine the value of $\underset{x\to 0}{\lim }\left[\frac{{\left(x^2{\mathrm{e}}^x-x^2\right)}^3}{x^9}\right]$.

Show that $P_i\left(n\right)=2n\text{sin}\left(\frac{\pi }{n}\right)$.

By writing $P_c\left(n\right)$ in the form $\frac{2\text{tan}\left(\frac{\pi }{n}\right)}{\frac{1}{n}}$, find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_c\left(n\right)$.

Find the perimeter of a regular hexagon, of side length, $x$ units, inscribed in a circle of radius 1 unit.

By finding a suitable number of derivatives of $f$, find the first two non-zero terms in the Maclaurin series for $f$.

Hence, write down the first four terms in what is called the Extended Binomial Theorem for ${\left(1+x\right)}^q\text{,}q\in \mathbb{Q}$.

Repeat this process to find the first four terms in a power series for ${\left(1+x\right)}^{-3}$.

By substituting $x=0$, find the value of $a_0$.

Use an appropriate Maclaurin series expansion to find $\underset{n\to \mathrm{\infty }}{\text{lim}}{P}_i\left(n\right)$ and interpret this result geometrically.

Using the result from part (c), find the Maclaurin series for $g\left(x\right)$ up to and including the $x^4$ term.

Use your answer to part (a)(i) to write down an estimate for $f\left(0.4\right)$.

Expand ${\left(1+x\right)}^5$ using the Binomial Theorem.

Consider a square of side length, $x$ units, inscribed in a circle of radius 1 unit. By dividing the inscribed square into four isosceles triangles, find the exact perimeter of the inscribed square.

The inequality found in part (h) can be used to determine lower and upper bound approximations for the value of $\pi$.

Determine the least value for $n$ such that the lower bound and upper bound approximations are both within 0.005 of $\pi$.

Let $g\left(x\right)={\text{e}}^{mx}, m\in \mathrm{\mathbb{Q}}$.

Consider the function $h$ defined by $h\left(x\right)=f\left(x\right)\times g\left(x\right)$ for $x>-1$.

It is given that the $x^2$ term in the Maclaurin series for $h\left(x\right)$ has a coefficient of $\frac{7}{4}$.

Find the possible values of $m$.

Use the Lagrange form of the error term to find an upper bound for the absolute value of the error in calculating $f(0.4)$, using the first three non-zero terms of the Maclaurin series for $f\left(x\right)$.

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

Consider an equilateral triangle ABC of side length, $x$ units, inscribed in a circle of radius 1 unit and centre O as shown in the following diagram.

The equilateral triangle ABC can be divided into three smaller isosceles triangles, each subtending an angle of $\frac{2\pi }{3}$ at O, as shown in the following diagram.

Using right-angled trigonometry or otherwise, show that the perimeter of the equilateral triangle ABC is equal to $3\sqrt{3}$ units.

Use the Maclaurin series for $\ln (1+x)$ to write down the first three non-zero terms of the Maclaurin series for $f\left(x\right)$.

Show that $P_c\left(n\right)=2n\text{tan}\left(\frac{\pi }{n}\right)$.

Hence find the first three non-zero terms of the Maclaurin series for $\frac{x}{1+x^2}$.

Hence, using integration, find the power series for $\text{arctan}x$, giving the first four non-zero terms.

Repeat this procedure to find $a_2$ and $a_3$.

Find the value of $a$ and the value of $b$.

Prove by mathematical induction that $\frac{d^n}{d{x}^n}\left(x^2{\mathrm{e}}^x\right)=\left[x^2+2nx+n\left(n-1\right)\right]{\mathrm{e}}^x$ for $n\in {\mathrm{\mathbb{Z}}}^{+}$.

Show that the coefficient of $x^3$ in the Maclaurin series for $f\left(x\right)$ is zero.

Differentiate the equation obtained part (b) and hence, find the first four terms in a power series for ${\left(1+x\right)}^{-2}$.

Use the results from part (d) and part (f) to determine an inequality for the value of $\pi$ in terms of $n$.

Find the first two derivatives of $f\left(x\right)$ and hence find the Maclaurin series for $f\left(x\right)$ up to and including the $x^2$ term.

Hence, by recognising the pattern, deduce the first four terms in a power series for ${\left(1+x\right)}^{-n}$, $n\in {\mathbb{Z}}^{+}$.

Using the Maclaurin series for $\text{arctan}x$ and ${\text{e}}^{3x}-1$, find the Maclaurin series for $\text{arctan}\left({\text{e}}^{3x}-1\right)$ up to and including the $x^3$ term.

Hence, or otherwise, find $\underset{x\to 0}{\text{lim}}\frac{f\left(x\right)-1}{\text{arctan}\left({\text{e}}^{3x}-1\right)}$.

By differentiating both sides of the expression and then substituting $x=0$, find the value of $a_1$.

With reference to the Lagrange form of the error term, explain whether your answer to part (b) is an overestimate or an underestimate for $f(0.4)$.

Consider the power series $1-x+x^2-x^3+x^4-...$

By considering the ratio of consecutive terms, explain why this series is equal to ${\left(1+x\right)}^{-1}$ and state the values of $x$ for which this equality is true.

Write down the power series for $\frac{1}{1+x^2}$.

By using the Maclaurin series for $\cos x$ and the result from part (a), show that the Maclaurin series for $\text{sec} x$ up to and including the term in $x^4$ is $1+\frac{x^2}{2}+\frac{5x^4}{24}$.

By using the Maclaurin series for $\text{arctan} x$ and the result from part (b), find $\underset{x\to 0}{\lim }\left(\frac{x\text{\hspace{0.05em}arctan} 2x}{\text{sec} x-1}\right)$.

Hence or otherwise, determine the Maclaurin series of $f\left(x\right)=x^2{\mathrm{e}}^x$ in ascending powers of $x$, up to and including the term in $x^4$.

Hence determine the Maclaurin series for $\tan \left(\ln \right(1+x\left)\right)$ as far as the term in $x^3$.

State the restriction which must be placed on $x$ for this expansion to be valid.

Assuming the Maclaurin series for $\cos x$ and $\ln (1+x)$, show that the Maclaurin series for $\cos \left(\ln \right(1+x\left)\right)$ is

$1-\frac{1}{2}{x}^2+\frac{1}{2}{x}^3-\frac{5}{12}{x}^4+\dots$

By differentiating the series in part (a), show that the Maclaurin series for $\sin \left(\ln \right(1+x\left)\right)$ is $x-\frac{1}{2}{x}^2+\frac{1}{6}{x}^3+\dots$ .