Maclaurin series expansions: notes and practice questions

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

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

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

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

Show that $g''(x) = 2g'(x)$.

Hence, find the values of $g'''(0)$ and $g^{(4)}(0)$.

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

Hence, or otherwise, determine the value of $\lim_{x \to 0} \frac{2e^x \cosh x - 2 - 2x - 2x^2}{x^3}$.

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}(xe^{-x}) = (-1)^n (x-n)e^{-x}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, determine the Maclaurin series of $f(x) = xe^{-x}$ in ascending powers of $x$, up to and including the term in $x^4$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{(xe^{-x} - x)^2}{x^4}$.

The function $f$ is defined by $f(x) = (1-2x)^{-3}$. The first three terms of the Maclaurin series for $f(x)$ are $1 + ax + bx^2 + \dots$.

(a) (i) Show that $a=6$.

(ii) Find the value of $b$.

(b) The value of a rare collectible is modelled by the function $V(t) = 1000(1-0.05t)^{-3}$ dollars, where $t$ is the number of years from the start of 2024. Use the first three terms of the Maclaurin series for $f(x)$ to estimate the value of the collectible at the start of 2026.

Let $f(x) = \frac{1}{\sqrt{1-2x}}$ for $x < \frac{1}{2}$.

(a) Show that $f''(x) = 3(1-2x)^{-\frac{5}{2}}$.

(b) Use mathematical induction to prove that $f^{(n)}(x) = \frac{(2n)!}{2^n n!} (1-2x)^{-\frac{2n+1}{2}}$ for $n \in \mathbb{Z}, n \ge 2$.

Let $g(x) = \ln(1+kx)$, where $k$ is a real constant.

Consider the function $h$ defined by $h(x) = f(x) \times g(x)$ for $x < \frac{1}{2}$.

It is given that the coefficient of the $x^2$ term in the Maclaurin series for $h(x)$ is $-4$.

(c) Find the possible values of $k$.

Write down the first three terms of the binomial expansion of $(1-t)^{-2}$ in ascending powers of $t$.

The Maclaurin series for $\cos x$ is given by $\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots$

By using this series and the result from part (a), show that the Maclaurin series for $\sec^2 x$ up to and including the term in $x^4$ is $1 + x^2 + \frac{2x^4}{3}$.

The Maclaurin series for $\sin x$ is given by $\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots$

By using this series and the result from part (b), find

$\lim_{x \to 0} \frac{x \sin 3x}{\sec^2 x - 1}$.

Let $f(x) = \frac{1}{\sqrt{1+2x}}$ for $x > -\frac{1}{2}$.

(a) Show that $f''(x) = 3(1+2x)^{-\frac{5}{2}}$.

(b) Use mathematical induction to prove that $f^{(n)}(x) = (-1)^n \frac{(2n)!}{2^n n!} (1+2x)^{-\frac{2n+1}{2}}$ for $n \in \mathbb{Z}, n \ge 1$.

Let $g(x) = e^{mx}, m \in \mathbb{R}$.

Consider the function $h$ defined by $h(x) = f(x) \times g(x)$ for $x > -\frac{1}{2}$.

It is given that the $x^2$ term in the Maclaurin series for $h(x)$ has a coefficient of $3$.

(c) Find the possible values of $m$.

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}(xe^{-x}) = (-1)^n(x-n)e^{-x}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, find the Maclaurin series of $f(x) = xe^{-x}$ in ascending powers of $x$, up to and including the term in $x^5$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{xe^{-x} - x + x^2}{x^3}$.

(a) Prove by mathematical induction that $\frac{d^n}{dx^n}\left(\frac{x}{1-x}\right) = n!(1-x)^{-(n+1)}$ for $n \in \mathbb{Z}^+$.

(b) Hence or otherwise, determine the Maclaurin series of $f(x) = \frac{x}{1-x}$ in ascending powers of $x$, up to and including the term in $x^4$.

(c) Hence or otherwise, determine the value of $\lim_{x\to0} \frac{\left(\frac{x}{1-x} - x\right)^2}{x^4}$.

Consider the function $g(x)=\frac{1-\cos(ax)}{x^2}$, where $x \neq 0$ and $a \in \mathbb{R}^+$.

(a) Show that $g$ is an even function.

(b) Given that $\lim_{x\to0} g(x) = 8$, find the value of $a$.

Consider the function $g(x)=\frac{1-\cos(ax)}{e^{x^2}-1}$, where $x \neq 0$ and $a \in \mathbb{R}^+$.

(a) Show that $g$ is an even function.

(b) Given that $\lim_{x\to0} g(x) = 8$, find the value of $a$.

Find the first three non-zero terms in the Maclaurin series of $e^{-x^2}$.

Find the first three non-zero terms in the Maclaurin series of $e^{-2x^2}$.

Hence, or otherwise, find the first two non-zero terms in the Maclaurin series of $-4x e^{-2x^2}$.

A curve is given by the equation $e^y = \cos(x)$ for $x \in (-\frac{\pi}{2}, \frac{\pi}{2})$.

(a) Use implicit differentiation to show that $\frac{dy}{dx} = -\tan(x)$.

(b) Show that $\frac{d^2y}{dx^2} + (\frac{dy}{dx})^2 + 1 = 0$.

(c) Find an expression for $\frac{d^3y}{dx^3}$ in terms of $\frac{dy}{dx}$ and $\frac{d^2y}{dx^2}$.

(d) Hence, find the Maclaurin series for $y = \ln(\cos(x) )$ up to and including the term in $x^4$.

Let $f(x) = \ln(1+ax)$, where $ax > -1, a \neq 0$.

The $n^{\text{th}}$ derivative of $f(x)$ is denoted by $f^{(n)}(x)$, for $n \in \mathbb{Z}^+$.

Prove by induction that $f^{(n)}(x) = (-1)^{n-1} (n-1)! a^n (1+ax)^{-n}$, for $n \in \mathbb{Z}^+$.

Hence or otherwise, find the Maclaurin series for $f(x) = \ln(1+ax)$ up to and including the $x^3$ term.

Hence, find the series expansion of $\ln\left(\frac{1+2x}{1-3x}\right)$ up to and including the $x^3$ term.

State the restriction which must be placed on $x$ for the approximation in part (c) to be valid.

Use a suitable value of $x$ to determine an approximate value for $\ln(2)$.

Give your answer as a rational number.

Consider the family of integrals defined by $I_n = \int x^n e^{-x} \, dx$ for $n \in \mathbb{N}_0$, where $\mathbb{N}_0 = \{0, 1, 2, ...\}$.

(a) By using integration by parts, show that $I_n = -x^n e^{-x} + n I_{n-1}$ for $n \ge 1$.

(b) Hence, find an explicit expression for $\int x^3 e^{-x} \, dx$.

(c) The region $R$ is enclosed by the graph of $y = x^{3/2} e^{-x/2}$ and the $x$-axis for $x \ge 0$. The region $R$ is rotated by $2\pi$ radians about the $x$-axis. Find the volume of the solid generated.

(d) Show that $\lim_{x \to \infty} x^n e^{-x} = 0$ for any $n \in \mathbb{N}$.

Consider the function $h(x) = x e^{-x}$.

(e) (i) Find the Maclaurin series for $h(x)$ up to and including the term in $x^4$.

(ii) Hence, find the value of the fourth derivative of $h(x)$ at $x=0$, i.e. $h^{(4)}(0)$.

Let $f(x) = \frac{3x^2 + 5x + 1}{(1+x)^2(2-x)}$.

(a) Express $f(x)$ in the form $\frac{A}{1+x} + \frac{B}{(1+x)^2} + \frac{C}{2-x}$.

(b) Hence, find the Maclaurin series for $f(x)$ up to and including the term in $x^3$.

(c) State the interval of convergence for this series.

Consider the differential equation $\frac{dy}{dx} = y^2 \sin x$.

Given that $y = 1$ when $x = 0$, show that the solution to the equation is $y = \sec x$.

Determine the value of the constant $A$ for which the following limit exists, and evaluate the limit:

$\lim_{x\to 0} \frac{\sec x - A}{x^2}$

(a) (i) The function $f$ is defined by $f(x)=\frac{1}{1+x}$ where $-1<x<1$.

Use the Maclaurin series for $\frac{1}{1-u}$ to write down the first three non-zero terms of the Maclaurin series for $f(x)$.

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

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

(c) 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 $f(x)$.

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