Proof by mathematical induction: notes and practice questions

(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 $\lim_{x\to0} \frac{(xe^{-x} - x)^2}{x^4}$.

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 $k$.

Solve the inequality $4x^2 - 2x - 3 > 0$.

Use mathematical induction to prove that $3^n > 2n^2 + n$ for $n \in \mathbb{Z}, n \ge 3$.

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 a geometric sequence with first term 2 and common ratio 3.

$S_n$ is the sum of the first $n$ terms of the sequence.

(a) Find an expression for $S_n$, in the form $a^n+b$, where $a, b \in \mathbb{Z}$.

(b) Hence, show that $S_1 + S_2 + S_3 + \dots + S_n = \frac{3^{n+1}-2n-3}{2}$.

The following diagram shows the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$, with asymptotes at $y = -\frac{3\pi}{4}$ and $y = \frac{\pi}{4}$.

Describe a sequence of transformations that transforms the graph of $y = \arctan x$ to the graph of $y = \arctan\left(\frac{x-3}{2}\right) -\frac{\pi}{4}$ for $x \in \mathbb{R}$.

Show that $\arctan p + \arctan q = \arctan\left(\frac{p+q}{1-pq}\right)$ where $p, q > 0$ and $pq < 1$.

Using mathematical induction and the result from part (b), prove that

$\sum_{r=1}^{n} \arctan\left(\frac{1}{r^2+r+1}\right) = \arctan\left(\frac{n}{n+2}\right)$ for $n \in \mathbb{Z}^+$.

A student claims that $2^n > n^2$ for all integers $n \ge 5$.

(a) Show that $2n^2 > (n+1)^2$ for all integers $n \ge 3$.

(b) Use mathematical induction and the result from part (a) to prove that the student's claim is valid for all integers $n \ge 5$.

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.