MthT 430 Notes Chapter 5d Sequences and Limits

Sequences Cf. Spivak Chapter 22.

Definition. An infinite sequence is a function whose domain is N.

As a convention, we also allow the domain of a sequence to be a subset of N which includes all natural numbers sufficiently large .

Notation

If a is the name of the sequence, instead of listing the particular values by

a(1), a(2), …,

we almost always use the subscript notation

a1, a2, ….

We denote the sequence by

{an}

Limits of sequences

Definition. A sequence {a_n} converges to L (in symbols lim_{n → ∞}a_n = L) iff for every ϵ > 0, there is a natural number N such that, for all natural numbers n,

if n > N, then |an − L| < ϵ.

A sequence {a_n} is said to converge if it converges to L for some [finite!] number L, and to diverge if it does not converge.

Compare

•

For a function f whose domain includes all x sufficiently large and positive ,

lim x → ∞  f(x) = L.

•

For a sequence {a_n}, whose domain includes all n sufficiently large and positive ,

lim n → ∞  an = L.

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