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Convergence of Sequences

By Richard Baraniuk

Introduction

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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=( sinnπ2 cosnπ2 ) gn n 2 n 2

Example 3

A function sequence: g n t={1  if  0t<1n0  otherwise   g n t 1 0 t 1 n 0

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 gR g if for every ε>0 ε 0 there is an integer N N such that i ,iN:|gig|<ε i i N gi g ε We usually denote a limit by writing limit   i gi=g i gi g 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: limit   n gn=0 n gn 0 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 nN n N we have |gn0|=1n1N<ε gn 0 1 n 1 N ε Thus, limit   n gn=0 n gn 0

Example 5

Now let us look at the following non-convergent sequence gn={1  if  n=even-1  if  n=odd gn 1 n even -1 n odd This sequence oscillates between 1 and -1, so it will therefore never converge.