Math Problems Vault

Problem 2.195

Let $p_{1}, p_{2}, \ldots , p_{l}, a_{1}, a_{2}, \ldots , a_{l}$ be arbitrary positive numbers. Then

\lim _{n \to \infty } \sqrt[n]{p_{1}a_{1}^{n} + p_{2}a_{2}^{n} + \cdots + p_{l}a_{l}^{n}}

exists and it is equal to the largest among the numbers $a_{1}, a_{2}, \ldots , a_{l}$ .

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Problem 2.196

Under the same hypotheses as in Problem 195

\lim _{n \to \infty } \frac{p_{1}a_{1}^{n+1} + p_{2}a_{2}^{n+1} + \cdots + p_{l}a_{l}^{n+1}}{p_{1}a_{1}^{n} + p_{2}a_{2}^{n} + \cdots + p_{l}a_{l}^{n}} = \max (a_{1}, a_{2}, \ldots , a_{l}) .

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Problem 2.197

Let $f(x)$ be an arbitrary polynomial whose zeros are all real and positive and for which

-\frac{f'(x)}{f(x)} = c_{0} + c_{1}x + c_{2}x^{2} + \cdots + c_{n}x^{n} \cdots .

Show that

\lim _{n \to \infty } \frac{1}{\sqrt[n]{c_{n}}} = \lim _{n \to \infty } \frac{c_{n-1}}{c_{n}}

exists and that it is equal to the smallest zero of $f(x)$ .

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Problem 2.198

The two functions $\varphi (x)$ and $f(x)$ are continuous and positive on the interval $\left[a,b\right]$ . Then

\lim _{n \to \infty } \sqrt[n]{\int _{a}^{b} \varphi (x) \left[f(x)\right]^{n} \, dx}

exists and is equal to the maximum of $f(x)$ on $\left[a,b\right]$ .

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Problem 2.199

Under the same hypothesis as in Problem 198

\lim _{n \to \infty } \frac{\displaystyle \int _{a}^{b} \varphi (x) \left[f(x)\right]^{n+1} \, dx}{\displaystyle \int _{a}^{b} \varphi (x) \left[f(x)\right]^{n} \, dx} = \max f(x) .

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Problem 2.200

Let $k$ be a positive constant and $a < \xi < b$ . Show that for $a, b, \xi , k$ fixed and $n \to \infty$

\int _{a}^{b} e^{-kn(x-\xi )^{2}} \, dx \sim \sqrt{\frac{\pi }{kn}} .

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Problem 2.201

The functions $\varphi (x)$ , $h(x)$ and $f(x)=e^{h(x)}$ are defined on the finite or infinite interval $\left[a,b\right]$ and satisfy the following conditions:

(1) $\varphi (x)\left[f(x)\right]^{n} = \varphi (x)e^{nh(x)}$ is absolutely integrable over $\left[a,b\right]$ ; $n=0,1,2,\ldots$

(2) The function $h(x)$ attains its maximum only at the point $\xi$ in $(a,b)$ ; moreover, the least upper bound of $h(x)$ is smaller than $h(\xi )$ on any closed interval that does not contain $\xi$ ; there is, furthermore, a neighbourhood of $\xi$ where $h''(x)$ exists and is continuous; finally $h''(\xi ) < 0$ ,

(3) $\varphi (x)$ is continuous at $x=\xi$ , $\varphi (\xi ) \neq 0$ . Then the following asymptotic formula holds as $n \to \infty$

\int _{a}^{b} \varphi (x) \left[f(x)\right]^{n} \, dx \sim \varphi (\xi ) \left[f(\xi )\right]^{n+1/2} \sqrt{-\frac{2\pi }{nf''(\xi )}} = \varphi (\xi ) e^{nh(\xi )} \sqrt{-\frac{2\pi }{nh''(\xi )}} .

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Remark.

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We consider only a neighbourhood of $\xi$ and expand $h(x)$ in powers of $(x-\xi )$ up to terms of the second order.

Problem 2.202

Let $n$ be an integer, $n \to +\infty$ . Using the fact that

\int _{0}^{\frac{\pi }{2}} \sin ^{2n} x \, dx = \int _{0}^{\frac{\pi }{2}} \cos ^{2n} x \, dx = \frac{1 \cdot 3 \cdots (2n-1)}{2 \cdot 4 \cdots 2n} \cdot \frac{\pi }{2},

prove that

\frac{1 \cdot 3 \cdots (2n-1)}{2 \cdot 4 \cdots 2n} \sim \frac{1}{\sqrt{n\pi }} .

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Problem 2.203

We assume that $\lambda$ is real, $\lambda > 1$ ; $P_{n}(x)$ denotes the $n$ -th Legendre polynomial. As $n \to \infty$

P_{n}(\lambda ) \sim \frac{1}{\sqrt{2n\pi }} \frac{(\lambda +\sqrt{\lambda ^{2}-1})^{n+1/2}}{\sqrt[4]{\lambda ^{2}-1}} .

The positive value of the roots must be used.

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Remark.

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Problem 86 of Part VI.

Problem 2.204

The Bessel function $J_{v}(t)$ can be defined by Hansen's expansion

e^{it\cos x} = J_{0}(t) + 2\sum _{v=1}^{\infty } i^{v} J_{v}(t) \cos vx .

Derive the following asymptotic formula:

J_{v}(it) \sim i^{v} \frac{e^{t}}{\sqrt{2\pi t}}, \qquad t \to +\infty , \ v = 0, 1, 2, \ldots

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Problem 2.205

Show that for positive $n$ , $n \to +\infty$ ,

\Gamma (n+1) = \int _{0}^{\infty } e^{-x} x^{n} \, dx \sim \left(\frac{n}{e}\right)^{n} \sqrt{2\pi n}

and, more accurately,

\left(\frac{e}{n}\right)^{n} \Gamma (n+1) = \sqrt{2\pi n} + O\left(\frac{1}{\sqrt{n}}\right) .

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Remark.

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Problem 18 of this Part; Problem 167 of Part I.

Problem 2.206

Let $k$ and $l$ be real numbers, $k > 1$ . Prove that for $n \to +\infty$

\binom {nk+l}{n} \sim \frac{(k-1)^{n}}{\sqrt{2\pi n}} \left(\frac{k}{k-1}\right)^{nk+l+\frac{1}{2}} .

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Problem 2.207

Assume that $\alpha$ is real and that $t$ is positive and increases to infinity. Then

\int _{1}^{\infty } x^{\alpha } \left(\frac{te}{x}\right)^{x} \frac{dx}{x} \sim \sqrt{2\pi } \, t^{\alpha -\frac{1}{2}} e^{t} .

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Problem 2.208

Let $0 < \alpha < 1$ . The following approximation is justified as $\tau \to +0$ :

\int _{0}^{\infty } \exp \left(\frac{x^{\alpha }}{\alpha } - \tau x\right) \, dx \sim \sqrt{\frac{2\pi }{1-\alpha }} \, \tau ^{-\frac{\alpha }{2(1-\alpha )}-1} \exp \left(\frac{1-\alpha }{\alpha } \tau ^{-\frac{\alpha }{1-\alpha }}\right) .

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Problem 2.209

Let $\alpha > 0$ . As $t \to \infty$ we obtain

\int _{0}^{\infty } x^{-\alpha x} t^{x} \, dx \sim \sqrt{\frac{2\pi }{e\alpha }} \, t^{\frac{1}{2\alpha }} \exp \left(e^{-1} \alpha t^{\alpha }\right) .

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