Sheet 3 Moodle version

This is the same content as the pdf in the Sheet 3 assignment, but visible as a Moodle page.

Q1 

For each of the following choices of \( a_n \), and for arbitrary \( \epsilon > 0 \), find \( N \) such that \( |a_n| < \epsilon \) whenever \( n \geq N \).  [You need only to find a value of \( N \) that works, not necessarily the smallest such \( N \).]

1. \( \displaystyle \frac{1}{n^2+3}, \)
2. \( \displaystyle \frac{1}{n(n-\pi)}, \)
3. \( \displaystyle \frac{1}{\sqrt{5n-1}}. \)

Q2

Use sandwiching arguments to prove that, for each of the following choices of \( a_n \), the sequence \( (a_n) \) converges to 0:

1. \( \displaystyle \frac{n+1}{n^2+n+1},\)
2. \( \displaystyle 2^{-n}\cos(n^2), \)
3. \( \displaystyle \sin\frac{1}{n}, \)
4. \( \begin{cases} \displaystyle \frac{1}{2^n} & \textrm{if $n$ is prime},\\ \displaystyle -\frac{1}{3^n} & \textrm{otherwise}. \end{cases} \)

Q3

(a) Prove that \( \sqrt{n+1}-\sqrt{n} \rightarrow 0 \) by using the identity \( a - b = (\sqrt{a} -\sqrt{b})(\sqrt{a}+\sqrt{b}) \) (for \( a \), \( b \in \mathbb{R}^{\geq 0} \)).

(b) Prove that \( n^{1/n} \geq 1 \) for \( n = 1 \), \( 2 \), ....  Let \( a_n = n^{1/n} - 1 \).  Prove, by applying the binomial theorem to \( (1+a_n)^n \), that \[ a_n \leq \sqrt{\frac{2}{n-1}} \textrm{ for } n > 1. \]

Deduce that \( n^{1/n} \rightarrow 1 \).

Q4

(a) Write down, carefully quantified, what it means for a real sequence \( (a_n) \) (i) to be convergent; (ii) to be bounded; (iii) to tend to infinity.  Write down, carefully quantified, the negations of (i), (ii), (iii).

(b) Formulate and prove an analogue of the Sandwiching Lemma applicable to real sequences that tend to infinity.

(c) For each of the following choices of \( a_n \) decide whether or not the sequence tends to infinity:

1. \( \displaystyle \frac{n^2+n+1}{n+1}, \)
2. \( n^2 \sin n, \)
3. \( \displaystyle \frac{n^{3/4}}{\sqrt{5n-1}}, \)
4. \( \displaystyle \left(1 + \frac{1}{n}\right)^n. \)

Justify your answers briefly.

Q5

Assume that \( (a_n) \) is a sequence such that \( a_n \rightarrow 0 \).  Let \( (b_n) \) be a bounded sequence.  Prove that \( a_n b_n \rightarrow 0 \).

Give an example of a single sequence \( (a_n) \) such that \( a_n \rightarrow 0 \) and of appropriate sequences \( (c_n) \) to demonstrate that each of the following possibilities can occur:

1. \( a_n c_n \rightarrow 0 \) and \( (c_n) \) is unbounded;
2. \( a_n c_n \rightarrow \infty \);
3. \( (a_n c_n) \) converges to a non-zero limit;
   
4. \( (a_n c_n) \) is bounded and divergent;
5. \( a_n c_n \rightarrow -\infty \).

[The idea is to use the same sequence \( (a_n) \) for all of these parts, but you can use different \( (c_n) \) for each.]

Q6

For each of the following choices of \( z_n \), decide whether or not \( (z_n) \) converges:

1. \( \displaystyle \left(\frac{1}{1+\mathrm{i}}\right)^n,\)
2. \( \displaystyle \frac{(1-\mathrm{i})n}{n+\mathrm{i}}, \)
3. \( \displaystyle (-1)^n \frac{n+\mathrm{i}}{n}. \)

Give brief justifications for your answers.

Q7

(Later parts may be treated as optional.) Let \( c \) be a complex number. The complex numbers \( z_n(c) \) are defined recursively by

\[ z_1(c) = c, \quad z_{n+1}(c) = (z_n(c))^2 + c \textrm{ for } n \geq 1. \]

The Mandelbrot set is defined by

\[ M = \{ c \in \mathbb{C} : \textrm{ the sequence } (z_n(c)) \textrm{ is bounded}\}. \]

(a) Show that each of \( -2 \), \( -1 \), \( 0 \), \( \mathrm{i} \) lies in \( M \) but that \( 1 \not\in M \).

(b) Show that if \( c \in M \) then \( \overline{c} \in M \), where \( \overline{c} \) denotes the conjugate of \( c \).

(c) Show that if \( |c| \leq \frac{1}{4} \) then \( |z_n(c)| < \frac{1}{2} \) for all \( n \).  (So if \( |c| \leq \frac{1}{4} \) then \( c \in M \).)

(d) Show that if \( |c| = 2 + \epsilon \) where \( \epsilon > 0 \), then \( |z_n(c)| \geq 2 + a_n \epsilon \) for \( n \geq 1 \) where \( a_n = (4^n + 2)/6 \).  Deduce that the Mandelbrot set lies entirely within the disc \( |z| \leq 2 \).