Irrationality and transcendence proofs

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$'s and $-1$'s, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Prove that the sum of the series $\sum \varepsilon_{k} 2^{-n_{k}}$ is irrational if $\overline{\lim }\left(n_{k+1}-n_{k}\right)=+\infty$.

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$'s and $-1$'s, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\varlimsup \frac{n_{k+1}}{n_{1} n_{2} \ldots n_{k}}=+\infty$. Prove that

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$'s and $-1$'s, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\varlimsup \frac{\ln n_{k}}{2^{k}}=+\infty$. Prove that

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$'s and $-1$'s, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers.

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$'s and $-1$'s, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\varlimsup \frac{\ln n_{k}}{3^{k}}=+\infty$. Prove that

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$'s and $-1$'s. Prove the transcendence of the numbers $\sum \varepsilon_{k} 2^{-k!}$.

$\left\{ \varepsilon_{k}\right\}$ is an arbitrary sequence of $+1$'s and $-1$'s, and $\left\{ n_{k}\right\}$ is a strictly increasing sequence of natural numbers. Let $\overline{\lim } \frac{\ln n_{k}}{k!}>0$. Prove that

Irrationality of the number $e$. Using the expansions

prove that

Let $\sigma (n)$ be the sum of divisors of a natural number $n$. Prove the irrationality of the sums of the series

Irrationality of the number $\pi$.

Irrationality of the values of the exponential function at rational points.

Irrationality of the values of the tangent at rational points.

All three vertices of an equilateral triangle cannot be lattice points.

The decimal fraction

(formed by writing consecutively all the prime numbers) is irrational. [249, 110.]

The number

is irrational.

The number $e$ is not only irrational, but it is also not an algebraic number of the second degree, i.e. it cannot satisfy an equation of the form

where $a,b,c$ are integers and are not all $=0$.

If the Euler (Mascheroni) constant $C=-\Gamma '(1)$ were a rational number, then $\Gamma '(n+1)$ would have to be an integer for all integers $n$ from some point onward.

If both $\vartheta /\pi$ and $\cos \vartheta$ are rational, then $\cos \vartheta$ must have one of the following five values:

Consider the interior angle included between two contiguous faces of a regular solid and call it

according as the solid is a tetra-, hexa-, octa-, dodeca-, or icosahedron (a hexahedron is a cube). Show that the ratios

are irrational.

The numbers $\alpha_1,\alpha_2,\ldots ,\alpha_m$ are called linearly dependent if there exist rational integers $a_1,a_2,\ldots ,a_m$ not all zero such that $a_1\alpha_1+a_2\alpha_2+\cdots +a_m\alpha_m=0$. If no relation of this kind exists then $\alpha_1,\alpha_2,\ldots ,\alpha_m$ are called linearly independent. Thus, if $\beta \neq 0$, the numbers $\alpha$ and $\beta$ are linearly dependent or independent according as $\alpha /\beta$ is rational or irrational. The number $\alpha$ is algebraic if there exists an $n$ (a positive rational integer) such that $1,\alpha ,\alpha^2,\ldots ,\alpha^n$ are linearly dependent.

The numbers (angles) $T,H$ and $O$ are linearly dependent; in fact

(see 197.2; it can be also proved by elementary geometry). Yet $T$ and $O$ are linearly independent.

(continued). Show that $T,H,D$ and $I$ are linearly independent. Also $H,O,D$ and $I$ are linearly independent, and any linear homogeneous relation with rational coefficients between $T,H,O,D$ and $I$ can differ only trivially (that is, only by a numerical factor) from the one given in 197.3.

If both $\vartheta /\pi$ and $\tan^2\vartheta$ are rational then $\tan^2\vartheta$ must have one of the following five values (regarding $\infty =1/0$ as ``rational''):

With the exception of the four cases $n=1,2,4$, and 8, $\tan 2\pi /n$ is irrational for all positive rational integers $n$.

(continued). From this result derive 197.1 [38.1].

The number

is irrational.

The numbers $a$ and $b$ are positive; $a$ may be rational or irrational and the same holds for $b$, and so there are four possible cases. Show by means of examples that $a^b$ can be rational or irrational in each of these four cases (eight examples are required).

If $r$ is rational $(r \neq 0)$ and $x$ is irrational, prove that $r+x$ and $r x$ are irrational.

Prove that there is no rational number whose square is 12.