sequence

(noun)

an ordered list of objects

Related Terms

  • limit
  • Zeno's dichotomy

Examples of sequence in the following topics:

  • Sequences

    • A sequence is an ordered list of objects (or events).
    • Also, the sequence (1,1,2,3,5,8)(1, 1, 2, 3, 5, 8)(1,1,2,3,5,8), which contains the number 111 at two different positions, is a valid sequence.
    • Sequences can be finite, as in this example, or infinite, such as the sequence of all even positive integers (2,4,6,⋯)(2, 4, 6, \cdots)(2,4,6,⋯).
    • Finite sequences are sometimes known as strings or words, and infinite sequences as streams.
    • The empty sequence ()( \quad )() is included in most notions of sequence, but may be excluded depending on the context.
  • Summing an Infinite Series

    • A series is the sum of the terms of a sequence.
    • Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
    • The sequence of partial sums Sk{S_k}S​k​​ associated to a series ∑n=0∞an\sum_{n=0}^\infty a_n∑​n=0​∞​​a​n​​ is defined for each k as the sum of the sequence an{a_n}a​n​​ from a0a_0a​0​​ to aka_ka​k​​:
    • Infinite sequences and series can either converge or diverge.
    • An infinite sequence of real numbers shown in blue dots.
  • Series

    • A series is the sum of the terms of a sequence.
    • A series is, informally speaking, the sum of the terms of a sequence.
    • Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely.
    • Given an infinite sequence of numbers {an}\{ a_n \}{a​n​​}, a series is informally the result of adding all those terms together: a1+a2+a3+⋯a_1 + a_2 + a_3 + \cdotsa​1​​+a​2​​+a​3​​+⋯ .
    • The sequence of partial sums {Sk}\{S_k\}{S​k​​} associated to a series ∑n=0∞an\sum_{n=0}^\infty a_n∑​n=0​∞​​a​n​​ is defined for each k as the sum of the sequence {an}\{a_n\}{a​n​​} from a0a_0a​0​​ to aka_ka​k​​:
  • Alternating Series

    • Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
    • Proof: Suppose the sequence ana_na​n​​ converges to 000 and is monotone decreasing.
    • (The sequence {Sm}\{ S_m \}{S​m​​} is said to form a Cauchy sequence, meaning that elements of the sequence become arbitrarily close to each other as the sequence progresses.)
  • Convergence of Series with Positive Terms

    • For a sequence {an}\{a_n\}{a​n​​}, where ana_na​n​​ is a non-negative real number for every nnn, the sum ∑n=0∞an\sum_{n=0}^{\infty}a_n∑​n=0​∞​​a​n​​ can either converge or diverge to ∞\infty∞.
    • For a sequence {an}\{a_n\}{a​n​​}, where ana_na​n​​ is a non-negative real number for every nnn, the sequence of partial sums
    • Therefore, it follows that a series ∑n=0∞an\sum_{n=0}^{\infty} a_n∑​n=0​∞​​a​n​​ with non-negative terms converges if and only if the sequence SkS_kS​k​​ of partial sums is bounded.
  • Absolute Convergence and Ratio and Root Tests

    • In this example, the ratio of adjacent terms in the blue sequence converges to L=12L=\frac{1}{2}L=​2​​1​​.
    • Then the blue sequence is dominated by the red sequence for all n≥2n \geq 2n≥2.
    • The red sequence converges, so the blue sequence does as well.
  • Further Transcendental Functions

    • A transcendental function is a function that "transcends" algebra in the sense that it cannot be expressed in terms of a finite sequence of the algebraic operations of addition, multiplication, power, and root extraction.
    • Identify a transcendental function as one that cannot be expressed as the finite sequence of an algebraic operation
  • Comparison Tests

    • Since the sum of the sequence 1n\frac{1}{n}​n​​1​​ (i.e., ∑1n)\left ( \text{i.e., }\sum {\frac{1}{n}}\right)(i.e., ∑​n​​1​​) diverges, the limit convergence test tells that the original series (with n+12n2\frac{n+1}{2n^2}​2n​2​​​​n+1​​) also diverges.
  • The Integral Test and Estimates of Sums

    • The above examples involving the harmonic series raise the question of whether there are monotone sequences such that f(n)f(n)f(n) decreases to 000 faster than 1n\frac{1}{n}​n​​1​​but slower than 1n1+ε\frac{1}{n^{1 + \varepsilon}}​n​1+ε​​​​1​​ in the sense that:
    • Once such a sequence is found, a similar question can be asked of f(n)f(n)f(n) taking the role of 1n\frac{1}{n}​n​​1​​ oand so on.
  • Limits and Continuity

    • However, fff itself is not continuous as can be seen by considering the sequence f(1n,1n)f \left(\frac{1}{n},\frac{1}{n} \right)f(​n​​1​​,​n​​1​​) (for natural nnn), which should converge to f(0,0)=0\displaystyle{f (0,0) = 0}f(0,0)=0 if fff is continuous.
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