Math Problems Vault

Problem 1.156

Assume that $\varphi (x)$ is defined for positive $x$ and that for $x$ large enough it can be represented in the form

\varphi (x) = a_0 + \frac{a_1}{x} + \frac{a_2}{x^2} + \cdots + \frac{a_k}{x^k} + \cdots

$a_k$ real, $k=0,1,2,\ldots$ The infinite series

\varphi (1)+\varphi (2)+\varphi (3)+\cdots +\varphi (n)+\cdots

converges if and only if $a_0=a_1=0$ .

?

Problem 1.157

(continued). Suppose $\varphi (n) \neq 0$ . The infinite product

\varphi (1)\varphi (2)\varphi (3) \cdots \varphi (n) \cdots

converges if and only if $a_0=1$ , $a_1=0$ .

?

Problem 1.158

(continued). Under what conditions does the following infinite series converge

\varphi (1) + \varphi (1)\varphi (2) + \varphi (1)\varphi (2)\varphi (3) + \cdots + \varphi (1)\varphi (2)\varphi (3) \cdots \varphi (n) + \cdots \, ?

?

Problem 1.159

For which positive values of $\alpha$ does the following series converge

\sum _{n=1}^{\infty } (2-e^{\alpha })(2-e^{\alpha /2}) \cdots (2-e^{\alpha /n}) \, ?

?

Problem 1.160

\int _0^1 x^{-x} \, dx = \sum _{n=1}^{\infty } n^{-n} .

?

Problem 1.161

Considering positive square roots only we find

\sqrt{1+\sqrt{1+\sqrt{1+\cdots }}} = 1 + \cfrac {1}{1+\cfrac {1}{1+\ddots }} \, .

?

Problem 1.162

Let $a_1, a_2, \ldots , a_n, \ldots$ be positive numbers and put

t_n = \sqrt{a_1+\sqrt{a_2+\cdots +\sqrt{a_n}}} \, ,

where only positive square roots are considered. The convergence of the sequence

(t) \qquad t_1, t_2, t_3, \ldots , t_n, \ldots

is related to

\limsup _{n \rightarrow \infty } \frac{\log \log a_n}{n} = \alpha

in the following way:

\text{if } \alpha <2 \text{ then } (t) \text{ converges,}

\text{if } \alpha >2 \text{ then } (t) \text{ diverges.}

?

Remark.

?

G. Pólya, Problem: Arch. Math. Phys. Ser. 3, Vol. 24, p. 84 (1916). Solved by G. Szegö: Arch. Math. Phys. Ser. 3, Vol. 25, pp. 88-89 (1917).

Problem 1.163

(continued). The sequence $(t)$ is certainly convergent if the series

\sum _{n=1}^{\infty } 2^{-n} a_n (a_1 a_2 \cdots a_n)^{-\frac{1}{2}} \text{ converges.}

?

Problem 1.164

We have for $0<q<1$

\frac{1-q}{1+q} \left(\frac{1-q^2}{1+q^2}\right)^{\frac{1}{2}} \left(\frac{1-q^4}{1+q^4}\right)^{\frac{1}{4}} \left(\frac{1-q^8}{1+q^8}\right)^{\frac{1}{8}} \cdots = (1-q)^2 .

?

Remark.

?

Jacobi, l.c.solution 53, §52, Corollarium: Werke, Vol. 1, pp. 200-201.

Problem 1.165

The series

\cdots + f''(x) + f'(x) + f(x) + \int _0^x f(\xi _1) \, d\xi _1 + \int _0^x d\xi _1 \int _0^{\xi _1} f(\xi _2) \, d\xi _2 + \cdots

is infinite in two directions. Suppose it does converge uniformly on some interval. Which function does the series represent?

?

Problem 1.166

Let $\varphi_n(x)$ and $\psi_n(x)$ denote the polynomials of degree $n$ , $n=0,1,2,\ldots$ defined by the formulas

\varphi _0(x)=1, \quad \varphi _n'(x)=\varphi _{n-1}(x), \quad \varphi _n(0)=0 ,

\psi _0(x)=1, \quad \psi _n(x+1)-\psi _n(x)=\psi _{n-1}(x), \quad \psi _n(0)=0, \qquad n=1,2,3,\ldots

Find the two sums $\varphi (x)$ and $\psi (x)$ ,

\varphi (x) = \varphi _0(x)+\varphi _1(x)+\cdots +\varphi _n(x)+\cdots ,

\psi (x) = \psi _0(x)+\psi _1(x)+\cdots +\psi _n(x)+\cdots .

?

Problem 1.167

We define

x_n = y_n e^{-\frac{1}{12n}}, \qquad y_n = n! \, n^{-n-\frac{1}{2}} e^n, \qquad n=1,2,3,\ldots

Then each interval $(x_k,y_k)$ , $k=1,2,3,\ldots$ , contains the interval $(x_{k+1},y_{k+1})$ as a subinterval.

?

Problem 1.168

The sequence

a_n = \left(1+\frac{1}{n}\right)^{n+p}, \qquad n=1,2,3,\ldots

is monotone decreasing if and only if $p \geq \dfrac {1}{2}$ .

?

Remark.

?

I. Schur.

Problem 1.169

The sequence

a_n = \left(1+\frac{1}{n}\right)^n \left(1+\frac{x}{n}\right), \qquad n=1,2,3,\ldots

is monotone decreasing if and only if $x \geq \dfrac {1}{2}$ .

?

Problem 1.170

Let $n$ be a positive integer. Then we have

\frac{e}{2n+2} < e - \left(1+\frac{1}{n}\right)^n < \frac{e}{2n+1} .

?

Remark.

?

Cf.problem No. 1098, Nouv. Annls Math. Ser. 2, Vol. 11, p. 480 (1872). Solved by C. Moreau; Nouv. Annls Math. Ser. 2, Vol. 13, p. 61 (1874).

Problem 1.171

As is well-known, the number $e = \lim_{n \rightarrow \infty } \left(1+\dfrac {1}{n}\right)^n$ is contained in the interval

\left(1+\frac{1}{n}\right)^n < e < \left(1+\frac{1}{n}\right)^{n+1} \qquad \text{[problem 168]}

In which quarter of the interval is it contained?

?

Remark.

?

I. Schur.

Problem 1.172

The sequence

a_n = \left(1+\frac{x}{n}\right)^{n+1}, \qquad n=1,2,3,\ldots

is monotone decreasing if and only if $0<x \leq 2$ .

?

Remark.

?

I. Schur.

Problem 1.173

Prove that the $n$ -times iterated sine function

\sin _n x = \sin (\sin _{n-1} x), \qquad \sin _1 x = \sin x ,

converges to $0$ as $n \rightarrow \infty$ if $\sin x>0$ and that, moreover,

\lim _{n \rightarrow \infty } \sqrt{\frac{n}{3}} \sin _n x = 1 .

?

Remark.

?

Proof based on a communication of E. Jacobsthal.

Problem 1.174

Assume that $0<f(x)<x$ and

f(x) = x - ax^k + bx^l + x^l \varepsilon (x), \qquad \lim _{x \rightarrow 0} \varepsilon (x) = 0

for $0<x<x_0$ , where $1<k<l$ and $a,b$ positive. Put

v_0=x, \quad v_1=f(v_0), \quad v_2=f(v_1), \quad \ldots , \quad v_n=f(v_{n-1}), \quad \ldots

Then we have for $n \rightarrow \infty$

n^{\frac{1}{k-1}} v_n \rightarrow \big[(k-1)a\big]^{-\frac{1}{k-1}} .

?

Problem 1.175

Discuss the convergence of the series

v_1^s + v_2^s + v_3^s + \cdots ,

where

v_1 = \sin x, \quad v_2 = \sin \sin x, \quad \ldots , \quad v_n = \sin v_{n-1}, \quad \ldots

We can obviously assume that $v_1>0$ .

?

Remark.

?

J. Ouspensky, Problem: Arch. Math. Phys. Ser. 3, Vol. 20, p. 83 (1913).

Problem 1.176

Prove the formula

e^x - 1 = u_1 + u_1 u_2 + u_1 u_2 u_3 + \cdots

where

u_1 = x \geq 0, \qquad u_{n+1} = \log \frac{e^{u_n}-1}{u_n}, \qquad n=1,2,3,\ldots

?

Remark.

?

Cf.E. Cesàro, Problem: Nouv. Annls Math. Ser. 3, Vol. 7, p. 400 (1888). Solved by Audibert: Nouv. Annls Math. Ser. 3, Vol. 11, p. 35 (1892).

Problem 1.177

Compute

s = \cos ^3 \varphi - \frac{1}{3} \cos ^3 3\varphi + \frac{1}{3^2} \cos ^3 3^2\varphi - \frac{1}{3^3} \cos ^3 3^3 \varphi + \cdots .

?

Remark.

?

C. A. Laisant, Problem: Nouv. Annls Math. Ser. 2, Vol. 9, p. 144 (1870). Solved by H. Rumpen: Nouv. Annls Math. Ser. 2, Vol. 11, p. 232 (1872).

Problem 1.178

Let $a_n$ , $b_n$ , $b_n \neq 0$ , $n=0,1,2,3,\ldots$ , be two sequences that satisfy the conditions:

the power series $f(x) = \sum_{n=0}^{\infty } a_n x^n$ has a positive radius of convergence $r$ .
the limit
$\lim _{n \rightarrow \infty } \frac{b_n}{b_{n+1}} = q$
exists and $\left|q\right|<r$ .

We define

c_n = a_0 b_n + a_1 b_{n-1} + \cdots + a_n b_0, \qquad n=0,1,2,\ldots

Then $\dfrac {c_n}{b_n}$ converges to $f(q)$ as $n \rightarrow \infty$ .

?

Remark.

?

I. Schur, Problem: Arch. Math. Phys. Ser. 3, Vol. 27, p. 162 (1918). Cf.O. Szász: Sber. Berlin Math. Ges. Vol. 21, pp. 25-29 (1922).

Problem 1.179

Let

f_n(x) = a_{n1}x + a_{n2}x^2 + a_{n3}x^3 + \cdots , \qquad n=1,2,3,\ldots

be arbitrary functions, $\left|a_{nk}\right|<A$ for all positive integral values of $n$ and $k$ and

\lim _{n \rightarrow \infty } f_n(x) = 0 \quad \text{if} \quad 0<x<1 .

Then we have for fixed $k$ , $k=1,2,3,\ldots$

\lim _{n \rightarrow \infty } a_{nk} = 0 .

?

Remark.

?

Special case of an important proposition in function theory by Vitali. Cf.E. Lindelöf: Bull. Soc. Math. France Vol. 41, p. 171 (1913).

Problem 1.180

Suppose that the series

a_{n0}+a_{n1}+a_{n2}+\cdots +a_{nk}+\cdots = s_n, \qquad n=0,1,2,\ldots ,

have a common convergent majorant

A_0+A_1+A_2+\cdots +A_k+\cdots = S ,

i.e.for each $k$ the inequalities $\left|a_{nk}\right| \leq A_k$ hold simultaneously for all $n$ . Assume furthermore that

\lim _{n \rightarrow \infty } a_{nk} = a_k

exists for $k=0,1,2,\ldots$ Then the series

a_0+a_1+a_2+\cdots +a_k+\cdots = s

is convergent and

\lim _{n \rightarrow \infty } s_n = s .

?

Problem 1.181

Justify the limits in problems 53 and 59.

?

Problem 1.181.1

Assume that

a_{i1}+a_{i2}+\cdots +a_{in}+\cdots = s_i

converges for $i=1,2,3,\ldots$ , define $U_i$ as the least upper bound of

\left|a_{i1}+a_{i2}+\cdots +a_{in}\right|, \qquad n=1,2,3,\ldots ,

and assume that

U_1+U_2+\cdots +U_n+\cdots

converges. Then the series

(*) \qquad a_{11}+a_{12}+a_{21}+a_{13}+a_{22}+a_{31}+\cdots +a_{1n}+a_{2,n-1}+\cdots ,

which you obtain by arranging the numbers in the array

\begin{matrix} a_{11} & a_{12} & a_{13} & \cdots & a_{1n} & \cdots \\ a_{21} & a_{22} & a_{23} & \cdots & a_{2,n-1} & \cdots \\ a_{31} & a_{32} & a_{33} & \cdots & & \\ \cdots & \cdots & \cdots & \cdots & \cdots & \end{matrix}

``diagonally'', converges and its sum is

s_1+s_2+\cdots +s_n+\cdots .

?

Remark.

?

The interesting point is that the absolute convergence of the double series $\sum \sum a_{ik}$ is not assumed.

Problem 1.182

If $\alpha$ is fixed, $\alpha >0$ , and $n$ an integer increasing to $+\infty$ , then

{\sum _{\nu =1}^{\infty }}' \, \nu ^{\alpha -1} (n^{\alpha }-\nu ^{\alpha })^{-2} \sim \frac{n^{1-\alpha }}{3} \left(\frac{\pi }{\alpha }\right)^2 .

The meaningless term with subscript $\nu =n$ has to be omitted, which is indicated by the comma at the summation sign. Notice the case $\alpha =1$ .

?

Problem 1.183

Let the numbers $\varepsilon_0, \varepsilon_1, \varepsilon_2, \ldots , \varepsilon_n, \ldots$ assume one of the three values $-1, 0, 1$ . Then we have

\varepsilon _0 \sqrt{2+\varepsilon _1 \sqrt{2+\varepsilon _2 \sqrt{2+\cdots }}} = 2 \sin \left( \frac{\pi }{4} \sum _{n=0}^{\infty } \frac{\varepsilon _0 \varepsilon _1 \varepsilon _2 \cdots \varepsilon _n}{2^n} \right) .

?

Remark.

?

The left hand side must be interpreted as the limit of

\varepsilon _0 \sqrt{2+\varepsilon _1 \sqrt{2+\varepsilon _2 \sqrt{2+\cdots +\varepsilon _n \sqrt{2}}}} \, , \qquad n=0,1,2,\ldots

for $n \rightarrow \infty$ . These expressions are well defined for all $n$ . Non-negative square roots are used throughout.

Problem 1.184

Every value $x$ of the interval $[-2,2]$ can be written in the form

x = \varepsilon _0 \sqrt{2+\varepsilon _1 \sqrt{2+\varepsilon _2 \sqrt{2+\cdots }}}

where $\varepsilon_k$ , $k=0,1,2,\ldots$ , is either $-1$ or $+1$ . The representation is unique if $x$ is not of the form $2 \cos \dfrac {p}{2^q} \pi$ , $p, q$ integers, $0<p<2^q$ . These are the only numbers that may be written in the finite form

\varepsilon _0 \sqrt{2+\varepsilon _1 \sqrt{2+\varepsilon _2 \sqrt{2+\cdots +\varepsilon _n \sqrt{2}}}} \, .

The finite representation can be extended to an infinite one in two ways: by putting either

\varepsilon _{n+1}=1, \quad \varepsilon _{n+2}=-1, \quad \varepsilon _{n+3}=\varepsilon _{n+4}=\cdots =1 ,

or

\varepsilon _{n+1}=-1, \quad \varepsilon _{n+2}=-1, \quad \varepsilon _{n+3}=\varepsilon _{n+4}=\cdots =1 .

?

Remark.

?

Cf.S. Pincherle: Atti Accad. Sci. Torino, Cl. Sci. Fis. Mat. Natur. Vol. 53, pp. 745-763 (1917-1918); Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. Ser. 5, Vol. 27, 2nd Sem. pp. 177-183 (1918).

Problem 1.185

(continued). The number $x$ is of the form $x=2\cos k\pi$ , $k$ rational, if and only if the sequence $\varepsilon_0, \varepsilon_1, \varepsilon_2, \ldots$ is periodical after a certain term.

?

Problem 1.185.1

Construct a sequence of real numbers $a_1, a_2, \ldots , a_n, \ldots$ so that the series

a_1^l + a_2^l + \cdots + a_n^l + \cdots

diverges for $l=5$ , but converges when $l$ is any odd positive integer different from 5.

?

Remark.

?

For precursory heuristic considerations see MD, Vol. 2, pp. 49-50 and 171.

Problem 1.185.2

(continued). Let the set of all odd positive integers be divided (arbitrarily) into two complementary subsets $C$ and $D$ (having no element in common). Show that there exists a sequence of real numbers $a_1, a_2, \ldots , a_n, \ldots$ such that the series mentioned converges when $l$ belongs to $C$ and diverges when $l$ belongs to $D$ .

?

Remark.

?

G. Pólya, Problem: Amer. Math. Monthly Vol. 51, p. 593 (1944). Solved by N. J. Fine: Amer. Math. Monthly Vol. 53, pp. 283-284 (1946).