Math Problems Vault

Problem 2.30

Assume that the monotone function $f(x)$ is defined for $x \geq 0$ and that the improper integral $\int_{0}^{\infty } f(x) \, dx$ exists. Then we have

\lim _{h \to +0} h \left(f(h) + f(2h) + f(3h) + \cdots \right) = \lim _{h \to +0} h \sum _{n=1}^{\infty } f(nh) = \int _{0}^{\infty } f(x) \, dx .

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Problem 2.31

The $\Gamma$ -function is defined for $\alpha > 0$ (or $\Re \alpha > 0$ ) by the integral

\Gamma (\alpha ) = \int _{0}^{\infty } e^{-x} x^{\alpha -1} \, dx .

Using Problem 89 of Part I prove that

\Gamma (\alpha ) = \lim _{n \to \infty } \frac{n^{\alpha -1} n!}{\alpha (\alpha +1) \cdots (\alpha +n-1)}, \qquad \alpha > 0 .

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Problem 2.32

As it is well known [cf. E. T. Whittaker and G. N. Watson, p. 246] Euler's constant $C$ can be written as an integral

C = \int _{0}^{\infty } e^{-x} \left(\frac{1}{1 - e^{-x}} - \frac{1}{x}\right) \, dx .

Show that

C = \lim _{t \to 1-0} \left[(1-t) \left(\frac{t}{1-t} + \frac{t^{2}}{1-t^{2}} + \frac{t^{3}}{1-t^{3}} + \cdots + \frac{t^{n}}{1-t^{n}} + \cdots \right) - \log \frac{1}{1-t}\right] .

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Problem 2.33

\lim _{t \to 1-0} (1-t) \left(\frac{t}{1+t} + \frac{t^{2}}{1+t^{2}} + \frac{t^{3}}{1+t^{3}} + \cdots + \frac{t^{n}}{1+t^{n}} + \cdots \right) = \log 2 .

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Problem 2.34

\lim _{t \to 1-0} (1-t)^{2} \left(\frac{t}{1-t} + 2 \frac{t^{2}}{1-t^{2}} + 3 \frac{t^{3}}{1-t^{3}} + \cdots + n \frac{t^{n}}{1-t^{n}} + \cdots \right) = \frac{\pi ^{2}}{6} .

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Problem 2.35

We have

\lim _{t \to 1-0} \sqrt{1-t} \, (1 + t + t^{4} + t^{9} + \cdots + t^{n^{2}} + \cdots ) = \frac{\sqrt{\pi }}{2}

and more generally for $\alpha > 0$

\lim _{t \to 1-0} \sqrt[\alpha ]{1-t} \, (1 + t^{1^{\alpha }} + t^{2^{\alpha }} + t^{3^{\alpha }} + \cdots + t^{n^{\alpha }} + \cdots ) = \frac{1}{\alpha } \Gamma \left(\frac{1}{\alpha }\right) .

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Problem 2.36

Compute

\lim _{t \to +\infty } \left(\frac{1}{t} + \frac{2t}{t^{2}+1^{2}} + \frac{2t}{t^{2}+2^{2}} + \cdots + \frac{2t}{t^{2}+n^{2}} + \cdots \right) .

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Problem 2.37

Let $\alpha > 1$ and put $g(t) = \prod_{n=1}^{\infty } \left(1 + \dfrac {t}{n^{\alpha }}\right)$ . Show that

\lim _{t \to +\infty } \frac{\log g(t)}{t^{\frac{1}{\alpha }}} = \frac{\pi }{\sin \dfrac {\pi }{\alpha }} .

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Problem 2.38

Establish the equation

\int _{0}^{\infty } \log (1 - 2x^{-2} \cos 2\varphi + x^{-4}) \, dx = 2\pi \sin \varphi , \qquad 0 \leq \varphi \leq \pi ,

with help of the following identity, valid for all complex $t$ :

\frac{\sin t}{t} = \frac{e^{it} - e^{-it}}{2it} = \prod _{n=1}^{\infty } \left(1 - \frac{t^{2}}{n^{2}\pi ^{2}}\right) .

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Problem 2.39

Compute the integral $\int_{0}^{a} \log x \, dx$ as the limit of an infinite sum of rectangles corresponding to the points of division

a, aq, aq^{2}, aq^{3}, \ldots , \qquad 0 < q < 1 .

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Problem 2.40

Let $k$ be a fixed positive number and $n$ an integer increasing to infinity. Then

\sum _{v=0}^{n} \binom {n}{v}^{k} \sim \frac{2^{kn}}{\sqrt{k}} \left(\frac{2}{\pi n}\right)^{\frac{k-1}{2}}

(Problem 58). Observe the particular cases $k = 1$ , $k = 2$ .

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