Winter 2016
MATH 226: Limits and infinite series
Branko Ćurgus
 Friday, February 12, 2016

 Today in class I handed out the
new assignment which is due next Friday.
 Thursday, February 11, 2016

 Theorem. Let $a : \mathbb N \to \mathbb R$ and $b : \mathbb N \to \mathbb R$ be two sequences and $K, L \in \mathbb R$.
Assume:

$\displaystyle \lim_{n\to+\infty} a_n = K$.

$\displaystyle \lim_{n\to+\infty} b_n = L$.

There exists $n_0 \in \mathbb N$ such that for all $n \in \mathbb N$ such that $n \geq n_0$ we have $a_n \leq b_n$.
Then $K \leq L$.
 Proof. Assume that the conditions 1, 2 and 3 in the theorem are satisfied.
Let $\epsilon \gt 0$ be arbitrary.
By definition of convergence for sequences the condition 1. implies that there exists $N_a(\epsilon) \in \mathbb R$ such that
\[
n \in \mathbb N \quad \text{and} \quad n \gt N_a(\epsilon) \quad \Rightarrow \quad a_n  K  \lt \epsilon.
\]
Notice that the condition $ a_n  K  \lt \epsilon$ is equivalent to $K  \epsilon \lt a_n \lt K + \epsilon$. Therefore the last displayed implication can be rewritten as
\[ \tag{G1}
n \in \mathbb N \quad \text{and} \quad n \gt N_a(\epsilon) \quad \Rightarrow \quad K  \epsilon \lt a_n \lt K + \epsilon.
\]
Similarly, by definition of convergence for sequences the condition 2. implies that there exists $N_b(\epsilon) \in \mathbb R$ such that
\[ \tag{G2}
n \in \mathbb N \quad \text{and} \quad n \gt N_b(\epsilon) \quad \Rightarrow \quad L  \epsilon \lt b_n \lt L + \epsilon.
\]
Let $m \in \mathbb N$ be such that $m \gt \max\bigl\{n_0, N_a(\epsilon), N_b(\epsilon)\bigr\}$.
From (G1), the condition 3. and (G2) we deduce that
\begin{align*}
K \epsilon \lt & \ a_m \lt K+\epsilon \\
& \ a_m \leq b_m \\
L &  \epsilon \lt b_m \lt L+\epsilon.
\end{align*}
From the last three displayed relations we deduce that
\[
K \epsilon \lt a_m \leq b_m \lt L+\epsilon.
\]
Consequently,
\[
K  L \lt 2 \epsilon.
\]
Now recall that $\epsilon \gt 0$ was arbitrary. Since the inequality $K  L \lt 2 \epsilon$ holds for all $\epsilon \gt 0$ we conclude that $K  L \leq 0$. Hence $K \leq L$ and this completes the proof.
 Friday, January 22, 2016


In this file I summarize steps involved in limit proofs for .

Do limits in Exercises 4.10.

Pay special attention to limits of specific functions in which the real number $a$ is not specified. For example, in class we proved
\[
\forall \ a \gt 0 \qquad \lim_{x\to a} \frac{1}{x} = \frac{1}{a} \qquad \text{and} \qquad \forall \ a \in \mathbb R \qquad \lim_{x\to a} x^2 = a^2.
\]

Do Exercise 4.11, Exercise 4.12 and limits in Exercises 4.15.
 Monday, January 11, 2016

 In
this file I summarize steps involved in limit proofs for .
 As a very simple example how to use Mathematica
here is the file that I created today.
 Another example of a Mathematica notebook is
this Mathematica file. In this file I show how to explore functions in Mathematica. The file is called PlottingFunctions.nb. Rightclick on the underlined word "Here"; in the popup menu that appears, your browser will offer you to save the file in your directory. Make sure that you save it with the exactly same name.

After saving the file you can open it with Mathematica. For this file use Mathematica 5.2. (We also have Mathematica 8 in BH 215. These two versions are not compatible.) You will find Mathematica 5.2 on most Windows computers on campus; click here for a list of labs with Mathematica installed (select Mathematica from the long list of programs and click search). To locate Mathematica on a particular computer you might try
Start > All Programs > Math Applications > Mathematica.
Open Mathematica first; then open PlottingFunctions.nb from Mathematica. You can execute the entire file by the following manu sequence (in Mathematica):
Kernel > Evaluation > Evaluate Notebook.
There are some more instructions in the file.

To get started with Mathematica 5.2 see my
Mathematica page.
 Tuesday, January 5, 2016
