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Convergence of Sequences

Module by: Richard Baraniuk

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

What is a Sequence?

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=sinnπ2cosnπ2 gn n 2 n 2

Example

A function sequence: g n t=1if0≤t<1n0otherwise 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 limi→∞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 1

We are given the following convergent sequence:

gn=1n gn 1 n (1)
Intuitively we can assume the following limit: limn→∞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 n≥N n N we have |gn-0|=1n≤1N<ε gn 0 1 n 1 N ε Thus, limn→∞gn=0 n gn 0

Example 2

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

Problems

For practice, say which of the following sequences converge and give their limits if they exist.

  1. gn=n gn n
  2. gn= 1nifn=even-1nifn=odd gn 1 n n even -1 n n odd
  3. gn= 1nifn≠power of 101otherwise gn 1 n n power of 10 1
  4. gn=nifn<1051nifn≥105 gn n n 10 5 1 n n 10 5
  5. gn=sinπn gn n
  6. gn=ⅈn gn n

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