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Special Limits

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Limits, Series, and Fractional Part Integrals

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Abstract

One says that a quantity is the limit of another quantity if the second approaches the first closer than any given quantity, however small…

The last thing one knows when writing a book is what to put first.

Blaise Pascal (1623–1662)

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Notes

  1. 1.

    A measurable function φ(t) on ℝ islocally integrable, φ ∈ L loc 1, if | φ | is integrable over any compact set (see [56, p. 215]).

  2. 2.

    The problem is motivated by the following limit of Traian Lalescu (see [82])

    $$\displaystyle{\lim \limits _{n\rightarrow \infty }\left (\sqrt[n + 1](n + 1)! -\sqrt[n]n!\right ) = 1/e.}$$
  3. 3.

    The Euler–Mascheroni constant, γ, is defined by γ = lim n → ∞ (1 + 1 ∕ 2 + ⋯ + 1 ∕ n − lnn). 

  4. 4.

    This improper integral is attributed to the Swiss mathematician and physicist J.L. Raabe (1801–1859). We learned from the work of Ranjan Roy [110, Exercise 2, p. 489] that an interesting proof of this integral, involving Euler’s reflection formula Γ(x)Γ(1 − x) = π ∕ sin(π x), was published by Stieltjes in 1878 (see [125, pp. 114–118]). Stieltjes has also proved that for x ≥ 0 one has

    $$\displaystyle{\int _{0}^{1}\ln \mathrm{\Gamma }(x + u)\mathrm{d}u = x\ln x - x + \frac{1} {2}\ln (2\pi ).}$$
  5. 5.

    This part of the proof is due toManyama (see [87]).

  6. 6.

    Hölder’s inequality states that if p > 1 and q are positive real numbers such that 1 ∕ p + 1 ∕ q = 1 and f and g are functions defined on [a, b], then

    $$\displaystyle{\int _{a}^{b}\vert f(x)g(x)\vert \mathrm{d}x \leq {\left (\int _{a}^{b}\vert f(x){\vert }^{p}\mathrm{d}x\right )}^{1/p}{\left (\int _{a}^{b}\vert g(x){\vert }^{q}\mathrm{d}x\right )}^{1/q}.}$$

    Equality holds if and only if A | f(x) | p = B | g(x) | q for all x ∈ [a, b].

  7. 7.

    This is also the topic of a classical problem of Pólya and Szegö [104, Problem 198, p. 78]. It is worth mentioning that another closely related problem states that if f : [a, b] →  ℝ is integrable, then the function g : [1, ∞) →  ℝ defined by

    $$\displaystyle{g(p) ={ \left ( \frac{1} {b - a}\int _{a}^{b}\vert f(x){\vert }^{p}\mathrm{d}x\right )}^{1/p},}$$

    is monotone increasing.

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  1. Department of Mathematics, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

    Ovidiu Furdui

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Furdui, O. (2013). Special Limits. In: Limits, Series, and Fractional Part Integrals. Problem Books in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-6762-5_1

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