Note 8.1 - Introduction to Sequences
1 Introduction
In the three following notes, our goal is to approximate any smooth function into a function involving only addition and multiplication (i.e. polynomials). The approximation will be performed in ways so that the errors approach 0 as the degrees of polynomials approach ∞. In appropriate cases, these approximations work well with differentiations and integrations.
To conclude, we are turning smooth and continuous things into discrete things (ones we can count). Let us begin by studying countable things called sequences.
2 Definitions and Examples
Definition 2.1. A sequence of real numbers is a function f : N → R.
3 Convergence
We are often most interested in whether the list of number {an} approach some- thing as n → ∞. This is defined precisely by
Definition 3.1.
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4 Properties
The algebraic properties of lim hold for sequences:
For real sequence, there is very important characterization of convergent sequence. Let’s first define
Definition 4.1. A sequence {an} is bounded if there is M so that |an| ≤ M for all n.
Definition 4.2. A sequence is called monotonic if it is nondecreasing (an ≤ an+1for all n) or nonincreasing (an≥ an+1for all n).
The theorem is
Theorem 4.3. A bounded, monotonic sequence of real numbers is convergent.
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5 Calculus on Sequences
Clearly, we can not perform calculus on sequences as they are function defined on N with elements that are at least 1 unit apart. Nevertheless, If there exist a continuous or smooth function on R that covers {an}:
then many properties of f are true for {an}:
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6 Examples
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