3 Definition of the number e

Begin with L ^A TEX

Anna Lazi´ nska

October 2019

1 Introduction

First steps ...

After the vertical space, we create a short new paragraph with a horizontal space. Horizontal space.

New paragraph - let’s get started! We will observe the next line of this paragraph if we enter a longer sentence.

Another paragraph in this document.

To the right.

We can use different ways to justify text in one document. Now the paragraph is aligned to the right margin.

Then we return to the left. What’s next?

2 A bit of simple math

Mathematical formulas are placed between dollar characters (ie. $ formula $), when we stay in the line of text. If the formula is to be centered in a new line, we place its notation between pairs of dollar char- acters (ie. $$ formula $$) (note: the backslash character in the tex file is used so that the dollar character appears in the processed document)

Examples:

The quadratic equation x² − 5x − 6 = 0 has two real roots, and the equation

x²+ x + 7 = 0 has no real roots.

1

3 Definition of the number e

The numbers sequence of the form (1 + _n¹)ⁿ, n = 1, 2, . . . , is convergent. Its limit is denoted by e, so from the definition e = lim_n→∞(1 + _n¹)ⁿ.

Now in the middle of a new line e = lim

n→∞(1 + 1 n)ⁿ

and another way – with correspondingly large brackets e = lim

n→∞

 1 + 1

n

n

.

You can also put the last printout method in a line of text, ie. e = lim

n→∞ 1 + _n¹
n

. Exercise. Calculate the limit of the sequence (a_n) where a_n = 1 + ²_n
3n

, n ∈ N.

4 Numbered and bulleted lists

Selected issues regarding functions:

1. concept of a function;

1.1. field of a function;

1.2. range of a function;

2. ways of describing functions;

• description;

• table;

• graph;

• formula, e.g. f (x) = 1 + _x2^2x+4, x ∈ R;

3. equality of functions;

4. monotonicity of functions;

- constant functions;

- non-decreasing functions;

- non-increasing functions;

2

5. function composition;

6. function continuity.

In mathematical analysis, there are two equivalent definitions of the limit of a function at a point:

Cauchy - using the concept of neighborhood of the point;

Heine - based on sequences.

Now a task on mathematical logic using two-column notation.

Exercise. Check which of the following expressions are tautologies:

a) (p ∧ q) ⇒ p;

b) (p ⇒ q) ⇔ (∼ p ∨ q);

c) [(p ⇒ q) ∧ p] ⇒ q;

d) p ⇒ (∼ p ∨ q);

e) (p ⇒ q) ⇔ [(p ∧ q) ⇔ p];

f) (p ⇒ q) ⇒ [(p ∧ r) ⇒ q].

We return to a single column.

Contents

1 Introduction 1

2 A bit of simple math 1

3 Definition of the number e 2

4 Numbered and bulleted lists 2

3