Miscellaneous Problems

Suppose that $f(x)$ is a real function of the real variable $x$. If the functions $\left|f'(x)\right|, \left|f''(x)\right|, \ldots$ are steadily and strictly decreasing in the interval $[0,x]$, $x>0$, then $f(x)$ is enveloped in the strict sense by its Maclaurin series.

The functions

are strictly enveloped by their Maclaurin series for $x>0$.

(continued). Prove the same for the functions

(continued). Prove the same for the functions

Suppose that the terms of the series $a_0+a_1+a_2+\cdots$ are alternately positive and negative and that there exists a number $A$ such that

assumes always the same sign as the next term, $a_{n+1}$. Then the series envelops $A$ in the strict sense.

If the series $a_0+a_1+a_2+\cdots$, $a_n$ real, $n=0,1,2,\ldots$, envelops the real number $A$ and if in addition $\left|a_1\right|>\left|a_2\right|>\left|a_3\right|>\cdots$ then the terms $a_1,a_2,a_3,\ldots$ have alternating signs and $A$ is enveloped in the strict sense.

Let the function $f(x)$ assume real values for real $x$, $x>R>0$. If $f(x)$ is enveloped for $x>R$ by the real series $a_0+\dfrac {a_1}{x}+\dfrac {a_2}{x^2}+\dfrac {a_3}{x^3}+\cdots$ then the numbers $a_1,a_2,a_3,\ldots$ have alternating signs and the series is strictly enveloping.

Suppose that the real valued function $f(t)$ is infinitely often differentiable for $t \geq 0$ and that all its derivatives $f^{(n)}(t)$ ($n=0,1,2,\ldots$) have decreasing absolute values and converge to $0$ for $t \rightarrow \infty$. Then the integral

is, for real $x$, strictly enveloped by the series

The number $\dfrac {2}{3}$ is enveloped by the series

but not in the strict sense.

Plot the first seven terms of the series

successively as complex numbers and compute so the value of $e^i$ to three decimals.

Let $\mathfrak {H}$ denote a ray with origin $z=0$. Assume that along $\mathfrak {H}$ all the derivatives of the function $f(z)$ attain the maximum of their absolute values at the origin and only there; i.e.

whenever $z$ is on $\mathfrak {H}$ and $\left|z\right|>0$. Then:

1. The function $f(t)$ is enveloped for every $z$ on $\mathfrak {H}$ by the Maclaurin series

   $$f(0) + \frac{f'(0)}{1!} z + \frac{f''(0)}{2!} z^2 + \cdots \qquad \text{[problem 140].}$$

2. the function $F(z) = \displaystyle \int_0^{\infty } e^{-t} f\left(\frac{t}{z}\right) dt$ is for every $z$ on $\overline{\mathfrak {H}}$ enveloped by the series

   $$f(0) + \frac{f'(0)}{z} + \frac{f''(0)}{z^2} + \frac{f'''(0)}{z^3} + \cdots \qquad \text{[problem 147];}$$

   $\overline{\mathfrak {H}}$ is the mirror image of $\mathfrak {H}$ with respect to the $x$-axis; the integral is taken along the positive $t$-axis and converges under our assumptions on $f(z)$.

The Maclaurin series of $e^{-z}$, $\log (1+z)$ and $(1+z)^{-p}$, $p>0$, envelop the respective functions for $\Re z \geq 0$, $z \neq 0$.

Let $z$ be restricted by the following conditions:

Then the function $e^{z^2/2} \displaystyle \int_z^{\infty } e^{-t^2/2} \, dt$ is enveloped by the series

(strictly enveloped if $z$ is real).

Suppose that $a_n$ and $b_n$ are arbitrary complex numbers that have the same argument, i.e.$\dfrac {a_n}{b_n}$ is real and positive. If at a certain point $z \neq 0$ the two series

envelop the values $\varphi (z)$ and $\psi (z)$ then the combined series

envelops the value $\varphi (z)+\psi (z)$ for this $z$. (The same is true for enveloping in the strict sense if all the coefficients and $z$ are real.)

If $z$ lies in the sectors described in problem 152 the function $z \coth z$ is enveloped by its power series

(strictly enveloped if $z$ is real). The coefficients $B_1, B_2, B_3, \ldots$ are called the Bernoulli numbers.

The function

can, for $\Re z>0$, be written as an integral

[Cf.E.T.Whittaker and G.N.Watson, pp.251-252.] Prove that the resulting (divergent) Stirling series

envelops the function $\omega (z)$ if $\Re z>0$ and $-\dfrac {\pi }{4} \leq \arg z \leq \dfrac {\pi }{4}$.

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

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

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

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

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

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

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

Considering positive square roots only we find

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

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

is related to

in the following way:

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

We have for $0<q<1$

The series

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

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

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

We define

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

The sequence

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

The sequence

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

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

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

In which quarter of the interval is it contained?

The sequence

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

Prove that the $n$-times iterated sine function

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

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

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

Then we have for $n \rightarrow \infty$

Discuss the convergence of the series

where

We can obviously assume that $v_1>0$.

Prove the formula

where

Compute

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

1. the power series $f(x) = \sum_{n=0}^{\infty } a_n x^n$ has a positive radius of convergence $r$.

2. the limit

   $$\lim _{n \rightarrow \infty } \frac{b_n}{b_{n+1}} = q$$

   exists and $\left|q\right|<r$.

We define

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

Let

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

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

Suppose that the series

have a common convergent majorant

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

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

is convergent and

Justify the limits in problems 53 and 59.

Assume that

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

and assume that

converges. Then the series

which you obtain by arranging the numbers in the array

``diagonally'', converges and its sum is

Assume that

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

and assume that

converges. Then the series

which you obtain by arranging the numbers in the array

``diagonally'', converges and its sum is

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

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

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

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

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

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

or

(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.

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

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

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

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

(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 $D$.

(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 $D$.

Tabulate the numbers $S_k^n$ for $n \leq 8$, $1 \leq k \leq n$.

Obviously

Show that

Show that

Show that, for $n \geq 1$,

Show that

Prove the identity in $x$

Prove the identity of problem 191 independently of problem 189 and derive hence a new proof for problem 189.

Define $T_0=1$. Then

Show that

Use problem 194 to prove problem 193 [see also problem 34].

Show that for $n \geq 1$

Tabulate the numbers $s_k^n$ for $n \leq 8$, $1 \leq k \leq n$.

Obviously

Show that

Show that

or, which is the same, that