Site Feedback

# Introduction

Insert paragraph text here.

# Sequences

Definition 1: sequence
A sequence is a function $gn gn$ defined on the positive integers '$nn$'. We often denote a sequence by $gn | n =1∞ n 1 gn$
Example 1

A real number sequence: $gn=1n gn 1 n$

Example 2

A vector sequence: $gn=( sinn⁢π2 cosn⁢π2 ) gn n 2 n 2$

Example 3

A function sequence:

Note:
A function can be thought of as an infinite dimensional vector where for each value of '$tt$' we have one dimension

# Convergence of Real Sequences

Definition 2: limit
A sequence $gn | n =1∞ n 1 gn$ converges to a limit $g∈R g$ if for every $ε>0 ε 0$ there is an integer $N N$ such that $∀ i ,i≥N:|gi−g|<ε i i N gi g ε$ We usually denote a limit by writing or $g i →g g i g$
The above definition means that no matter how small we make $εε$, except for a finite number of $g i g i$'s, all points of the sequence are within distance $εε$ of $gg$.

Example 4

We are given the following convergent sequence:

$gn=1n gn 1 n$
1
Intuitively we can assume the following limit: Let us prove this rigorously. Say that we are given a real number $ε>0 ε 0$. Let us choose $N=⌈1ε⌉ N 1 ε$, where $⌈x⌉ x$ denotes the smallest integer larger than $xx$. Then for $n≥N n N$ we have $|gn−0|=1n≤1N<ε gn 0 1 n 1 N ε$ Thus,

Example 5

Now let us look at the following non-convergent sequence This sequence oscillates between 1 and -1, so it will therefore never converge.