Cauchy sequence

(noun)

a sequence whose elements become arbitrarily close to each other as the sequence progresses

Related Terms

Examples of Cauchy sequence in the following topics:

  • Alternating Series

    • Like any series, an alternating series converges if and only if the associated sequence of partial sums converges.
    • Proof: Suppose the sequence $a_n$ converges to $0$ and is monotone decreasing.
    • (The sequence $\{ S_m \}$ is said to form a Cauchy sequence, meaning that elements of the sequence become arbitrarily close to each other as the sequence progresses.)

  • 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.
    • Infinite sequences and series can either converge or diverge.
    • An infinite sequence of real numbers shown in blue dots.
    • This sequence is neither increasing, nor decreasing, nor convergent, nor Cauchy.

  • The Integral Test and Estimates of Sums

    • It was developed by Colin Maclaurin and Augustin-Louis Cauchy and is sometimes known as the Maclaurin–Cauchy test.
    • The above examples involving the harmonic series raise the question of whether there are monotone sequences such that $f(n)$ decreases to $0$ faster than $\frac{1}{n}$but slower than $\frac{1}{n^{1 + \varepsilon}}$ in the sense that:
    • Once such a sequence is found, a similar question can be asked of $f(n)$ taking the role of $\frac{1}{n}$ oand so on.

  • Absolute Convergence and Ratio and Root Tests

    • The root test was developed first by Augustin-Louis Cauchy and so is sometimes known as the Cauchy root test, or Cauchy's radical test.
    • In this example, the ratio of adjacent terms in the blue sequence converges to $L=\frac{1}{2}$.
    • Then the blue sequence is dominated by the red sequence for all $n \geq 2$.
    • The red sequence converges, so the blue sequence does as well.

  • Sequences

    • A sequence is an ordered list of objects (or events).
    • Also, the sequence $(1, 1, 2, 3, 5, 8)$, which contains the number $1$ 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 $(2, 4, 6, \cdots)$.
    • 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.

  • Arithmetic Sequences

    • An arithmetic sequence is a sequence of numbers in which the difference between the consecutive terms is constant.
    • An arithmetic progression, or arithmetic sequence, is a sequence of numbers such that the difference between the consecutive terms is constant.
    • For instance, the sequence $5, 7, 9, 11, 13, \cdots$ is an arithmetic sequence with common difference of $2$.
    • The behavior of the arithmetic sequence depends on the common difference $d$.
    • Calculate the nth term of an arithmetic sequence and describe the properties of arithmetic sequences

  • DNA Sequencing Techniques

    • The Sanger sequencing method was used for the human genome sequencing project, which was finished its sequencing phase in 2003, but today both it and the Gilbert method have been largely replaced by better methods.
    • When the human genome was first sequenced using Sanger sequencing, it took several years, hundreds of labs working together, and a cost of around $100 million to sequence it to almost completion.
    • Sanger sequence can only produce several hundred nucleotides of sequence per reaction.
    • Most next-generation sequencing techniques generate even smaller blocks of sequence.
    • Most genomic sequencing projects today make use of an approach called whole genome shotgun sequencing.

  • Precise Definition of a Limit

    • It was first given by Bernard Bolzano in 1817, followed by a less precise form by Augustin-Louis Cauchy.
    • The letters $\varepsilon$ and $\delta$ can be understood as "error" and "distance," and in fact Cauchy used $\epsilon$ as an abbreviation for "error" in some of his work.

  • DNA Sequencing Based on Sanger Dideoxynucleotides

    • Sanger sequencing, also known as chain-termination sequencing, refers to a method of DNA sequencing developed by Frederick Sanger in 1977.
    • More recently, dye-terminator sequencing has been developed.
    • Automated DNA-sequencing instruments (DNA sequencers) can sequence up to 384 DNA samples in a single batch (run) in up to 24 runs a day.
    • Automation has lead to the sequencing of entire genomes.
    • Different types of Sanger sequencing, all of which depend on the sequence being stopped by a terminating dideoxynucleotide (black bars).

  • Strategies Used in Sequencing Projects

    • The strategies used for sequencing genomes include the Sanger method, shotgun sequencing, pairwise end, and next-generation sequencing.
    • All of the segments are then sequenced using the chain-sequencing method.
    • A larger sequence that is assembled from overlapping shorter sequences is called a contig.
    • This is the principle behind reconstructing entire DNA sequences using shotgun sequencing.
    • Compare the different strategies used for whole-genome sequencing:  Sanger method, shotgun sequencing, pairwise-end sequencing, and next-generation sequencing