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

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

Example

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

Example

A function sequence: g n ⁢t={1  if  0≤t<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 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 limit   i → ∞ gi=g i gi g or g i →g g i g