Processing math: 100%
Winter 2016
MATH 226: Limits and infinite series
Branko Ćurgus
- Friday, February 12, 2016
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- Today in class I handed out the
new assignment which is due next Friday.
- Thursday, February 11, 2016
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- 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:
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\displaystyle \lim_{n\to+\infty} a_n = K.
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\displaystyle \lim_{n\to+\infty} b_n = L.
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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
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In this file I summarize steps involved in limit proofs for .
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Do limits in Exercises 4.10.
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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.
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Do Exercise 4.11, Exercise 4.12 and limits in Exercises 4.15.
- Monday, January 11, 2016
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- 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. Right-click on the underlined word "Here"; in the pop-up 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.
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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.
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To get started with Mathematica 5.2 see my
Mathematica page.
- Tuesday, January 5, 2016
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