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  4. Convergence of sequences

16.1 Convergence of sequences

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This module will present an introduction into convergence and focus on what a sequence is and how it behaves as it approaches infinity.

Introduction

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Sequences

sequence
A sequence is a function g n defined on the positive integers ' n '. We often denote a sequence by n 1 ∞ g n

Convergence of real sequences

limit
A sequence n 1 ∞ g n converges to a limit g if for every ε 0 there is an integer N such that i i N g i g ε We usually denote a limit by writing i ∞ g i g or g i g
The above definition means that no matter how small we make ε , except for a finite number of g i 's, all points of the sequence are within distance ε of g .

We are given the following convergent sequence:

g n 1 n
Intuitively we can assume the following limit: n ∞ g n 0 Let us prove this rigorously. Say that we are given a real number ε 0 . Let us choose N 1 ε , where x denotes the smallest integer larger than x . Then for n N we have g n 0 1 n 1 N ε Thus, n ∞ g n 0

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

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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