620-295 Real Analysis with Applications

Assignment 3: Due 5pm on 4 September

Lecturer: Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville VIC 3010 Australia
aram@unimelb.edu.au

Due 5pm on 4 September in the appropriate assignment box on the ground floor of Richard Berry.

1. Define the following and give an example for each:

   (a)   cardinality,

   (b)   finite,

   (c)   infinite,

   (d)   countable,

   (e)   uncountable.

   
   
2. Prove that $\mathrm{Card}\left({\mathbb{Z}}_{>0}\right)\ne \mathrm{Card}\left(ℝ\right)$.
   
   
3. Define the following and give an example for each:

   (a)   sequence,

   (b)   converges (for a sequence),

   (c)   diverges (for a sequence),

   (d)   limit (of a sequence),

   (e)   sup (of a sequence),

   (f)   inf (of a sequence),

   (g)   lim sup (of a sequence),

   (h)   lim inf (of a sequence),

   (i)   bounded (for a sequence),

   (j)   increasing (for a sequence),

   (k)   decreasing (for a sequence),

   (l)   monotone (for a sequence),

   (m)   Cauchy sequence.

   
   
4. Give an example of a sequence $\left(a_n\right)$ such that none of $\inf a_n$, $\lim \inf a_n$, $\lim \sup a_n$, and $\sup a_n$ are equal.
   
   
5. Find the power series expansions and the radius of convergence of $e^x$, $\log (1+x)$, $\frac{1}{1-x}$, ${(1+x)}^{\mathrm{1/2}}$, $\arctan x$, and $\sinh x$.
   
   
6. Let $r\in ℝ$ with $0<r<1$. Find ${\displaystyle {\lim }_{n\to \infty }}{(1+\frac{\mathrm{r}}{n})}^n$, and explain why this limit is important to everyone with a credit card.
7. Prove that the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n}}$ diverges and that the series ${\displaystyle \sum _{n=1}^{\infty }\frac{1}{n^2}}$ converges.
   
   
8. Let $r\in ℝ$. Find (with proof) ${\displaystyle \sum _{n=1}^{\infty }{r}^n}$.
   
   
9. Show that the alternating harmonic series for $\arctan \mathrm{<mn>1</mn>}$ is conditionally convergent but not absolutely convergent. Explain how to rearrange it so that its sum is 301.