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MATH
[1251]

Definitions from Sequences and limits

All definitions from Sequences and limits.

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Real sequence

A real sequence is a function

a:N→R.a:\mathbb N\to\mathbb R.

Its value at nn is written ana_n and the sequence is written {an}\{a_n\}.

The starting index may be 00 or 11; changing finitely many initial terms does not change the limit. We will use the index that makes a given recurrence or series easiest to state.

Examples include

an=n2,bn=(−1)n,cn=1n+1,dn=e−n.a_n=n^2,\qquad b_n=(-1)^n,\qquad c_n=\frac1{n+1},\qquad d_n=e^{-n}.

A function of a real variable and a sequence are different objects: a sequence records only integer-indexed values, although a function ff can generate the sequence an=f(n)a_n=f(n). Sequences may also be recursive, as in

a1=a2=1,an+1=an+an−1(n≥2).a_1=a_2=1,\qquad a_{n+1}=a_n+a_{n-1}\quad(n\ge2).

Convergence of a sequence

The sequence {an}\{a_n\} converges to LL, written

lim⁡n→∞an=L,\lim_{n\to\infty}a_n=L,

if

∀ε>0 ∃N∈N such that n>N  ⟹  ∣an−L∣<ε.\forall\varepsilon>0\ \exists N\in\mathbb N \text{ such that }n>N\implies |a_n-L|<\varepsilon.

The index NN may depend on ε\varepsilon. The condition says that, for each ε\varepsilon, the tail eventually lies inside (L−ε,L+ε)(L-\varepsilon,L+\varepsilon). If no finite LL has this property, the sequence diverges.

Divergence includes several behaviours:

  • an→+∞a_n\to+\infty, as for n2n^2;
  • an→−∞a_n\to-\infty, as for −n-n;
  • bounded oscillation, as for sin⁡n\sin n;
  • unbounded oscillation, as for (−1)nn(-1)^n n.
  1. [1]:Infinite series
  2. [2]:Seen Questions
  3. [3]:Proving 2+1n+1→22+\frac1{n+1}\to2
  4. [4]:Algebra and continuity of sequence limits
  5. 1 more

Supremum and infimum

An upper bound of S⊆RS\subseteq\mathbb R is a number MM satisfying x≤Mx\le M for every x∈Sx\in S. The supremum sup⁡S\sup S is the least upper bound. Dually, a lower bound satisfies m≤xm\le x for every x∈Sx\in S and the infimum inf⁡S\inf S is the greatest lower bound.

The supremum or infimum need not belong to the set. For S=(0,1)S=(0,1), sup⁡S=1\sup S=1 and inf⁡S=0\inf S=0, although neither endpoint is in SS. This distinction will matter when the endpoints of a power-series interval are tested.

  1. [1]:Completeness of the real numbers

Monotone sequence

A sequence is non-decreasing if an+1≥ana_{n+1}\ge a_n for every nn and non-increasing if an+1≤ana_{n+1}\le a_n for every nn.

  1. [1]:Monotone convergence theorem

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Real sequenceConvergence of a sequenceSupremum and infimumMonotone sequence