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Monotone convergence

A sequence

x _ 1, x _ 2, x _ 3, ...(1)

is called

  • increasing if typeset structure for every typeset structure,
  • decreasing if typeset structure for every typeset structure,
  • non-increasing if typeset structure for every typeset structure,
  • non-decreasing if typeset structure for every typeset structure.

    • A sequence which is either increasing, decreasing, non-increasing, or non-decreasing is called a monotone sequence. A sequence which is either increasing, or  decreasing is called  strictly monotone.

    Theorem (The monotone convergence principle):  (a)  Let (1) be an increasing or non-decreasing sequence which is bounded above. Then (1) is a convergent sequence.
    (b) Let (1) be an decreasing or non-increasing sequence which is bounded below. Then (1) is a convergent sequence.
    (c) Let (1) be a monotone sequence which is bounded. Then (1) is a convergent sequence.

    Example: Let typeset structurebe fixed and

    s _ n = 1 + 1/2^p + 1/3^p + ... + 1/n^p                for       n = 1, 2, ... . (2)

    We have typeset structure for every typeset structure. We prove that if typeset structure then the sequence typeset structure, typeset structure,  is bounded above and thus convergent.  

    s _ n < s _ (2 n + 1) = 1 + (1/2^p + 1/3^p) + ... + (1/(2 n)^p + 1/(2 n + 1)^p) < 1 + 2/2^p + 2/4^p + ... + 2/(2 n)^p = 1 + 1/2^(p - 1) (1 + 1/2^p + ... + 1/n^p) = 1 + s _ n/2^(p - 1) .

    Consequently,

    s _ n(1 - 1/2^(p - 1)) < 1.

    If typeset structure we conclude that

    s _ n < 2^(p - 1)/(2^(p - 1) - 1)(3)

    for every typeset structure,

    Form estimation (3) there immediately follows that

    lim _ (n -> ∞) s _ n <= 2^(p - 1)/(2^(p - 1) - 1) .(4)

    [Graphics:HTMLFiles/MonotoneConvergence_23.gif]

    Cite this web-page as:

    Štefan Porubský: Monotone convergence.

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