Cauchy sequence

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.

• 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.

• Power Series

  • According to the Cauchy-Hadamard theorem, the radius $r$ can be computed from

• 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 $\{ a_n \}$, a series is informally the result of adding all those terms together: $a_1 + a_2 + a_3 + \cdots$ .
  • The sequence of partial sums $\{S_k\}$ associated to a series $\sum_{n=0}^\infty a_n$ is defined for each k as the sum of the sequence $\{a_n\}$ from $a_0$ to $a_k$:

• Convergence of Series with Positive Terms

  • For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sum $\sum_{n=0}^{\infty}a_n$ can either converge or diverge to $\infty$.
  • For a sequence $\{a_n\}$, where $a_n$ is a non-negative real number for every $n$, the sequence of partial sums
  • Therefore, it follows that a series $\sum_{n=0}^{\infty} a_n$ with non-negative terms converges if and only if the sequence $S_k$ of partial sums is bounded.

• 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