0
votes
1answer
19 views

Quotient of two rational sequences and the nature of its limit

Suppose we have two sequences of rational numbers, $(p_i)_{i=1}^\infty$ and $(q_i)_{i=1}^\infty$, and suppose $$\lim_{i\to\infty}\frac{p_i}{q_i}=c<\infty,$$ where $c$ is known. Are there any ...
0
votes
1answer
53 views

Monotone increasing sequence of rationals with an irrational limit

I am trying to use rationals in order to approximate irrationals. Is it possible to construct a monotonically increasing sequence of rationals the limit of which is an irrational? If so, how?
2
votes
1answer
26 views

Existence of a sequence of integers $\lbrace a_k\rbrace_{k\geq 1}$ so that the first $k$ digits of $a_k\alpha$ are $0$ where $\alpha$ is irrational.

Let $\alpha$ be an irrational number. Is there a sequence $\lbrace a_k\rbrace_{k\geq 1}$ of integers so that the first $k$ digits of the fractional part of $a_k·\alpha$ are $0$? (in base $2$, for ...
2
votes
4answers
135 views

For an irrational number $a$ the fractional part of $na$ for $n\in\mathbb N$ is dense in $[0,1]$

How to prove that the $\{$ fractional part of $n\alpha\mid n \in \mathbb{N}$ $\}$ is dense in $[0,1]$ for an irrational number $\alpha$. NOTICE that $n$ is in $\mathbb{N}$ Also notice that this is ...
1
vote
0answers
60 views

Irrational numbers and series

Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational ...
5
votes
1answer
125 views

If $(n_k)$ is strictly increasing and $\lim_{n \to \infty} n_k^{1/2^k} = \infty$ show that $\sum_{k=1}^{\infty} 1/n_k$ is irrational

Prove that for a strictly increasing natural sequence $(n_k) $ satisfying $\lim_{n \to \infty} n_k^{1/2^k}=\infty$, $\sum_{k=1}^{\infty} 1/n_k$ is irrational. This is another problem "problems in ...
7
votes
3answers
460 views

Sequences of Rationals and Irrationals

Let $(x_n)$ be a sequence that converges to the irrational number $x$. Must it be the case that $x_1, x_2, \dots$ are all irrational? Let $(y_n)$ be a sequences that converges to the rational number ...
2
votes
1answer
60 views

A series of reciprocal of integers [closed]

Let $F_n$ be integers, and $F_1<F_2<\cdots<F_n<\cdots$. Suppose that $$\lim_{n\to\infty}\frac{F_1F_2\cdots F_{n-1}}{F_n}=0.$$ Prove then $$\sum_{n=1}^\infty \frac{1}{F_n}$$ is convergence, ...
3
votes
2answers
335 views

Find the limiting value of the sequence

A sequence is given by the recurrence relation: $$u_n = 1 + {1\over u_{n-1} +1}, u_1 = 1, n{=\ge}1$$ Work out the 2nd, 3rd and 4th term of the sequence and find the limiting value of the sequence. ...
1
vote
2answers
79 views

Irrationality proof by rational approximations

Assume we have a sequence of rational numbers $\left(\frac{p_n}{q_n}\right),$ where $\gcd(p_n,q_n)=1, \ \forall n \in \mathbb N$. We know that $$\lim_{n\to\infty} \left(\frac{p_n}{q_n}\right)= x$$ ...
5
votes
1answer
145 views

If $q>1$ is not an integer, can $q^n$ be made arbitrarily close to integers?

This question arose when I heard about Mill's constant: the number $A$ such that $\lfloor A^{3^n} \rfloor$ is prime for all $n$. It made me wonder whether $A^{3^n}$ could be made arbitrarily close to ...
7
votes
4answers
1k views

Are irrational numbers completely random?

As far as I know the decimal numbers in any irrational appear randomly showing no pattern. Hence it may not be possible to predict the $n^{th}$ decimal point without any calculations. So I was ...
14
votes
2answers
416 views

What is the value of $\sum\limits_{i=1}^\infty\frac{1}{p_{p_i}}$ where $p_{i}$ is the $i$th prime?

What is the value of $\sum\limits_{i=1}^\infty\dfrac{1}{p_{p_i}}$ where $p_n$ is the nth prime (and so $p_{p_n}$ is the $k$th prime, where $k$ is the $n$th prime) ? Thus ...
3
votes
3answers
120 views

More three-term arithmetic progression questions

This question was inspired by a recent question about whether $\frac1{2}$, $\frac1{3}$ and $\frac1{15}$ can be (possibly non-consecutive) terms in an arithmetic progression. My question(s): Which of ...
2
votes
2answers
82 views

Could you prove convergence?

Two questions: Prove $a_n$ is bounded by 2 if $a_1=0, a_2=\sqrt2,\ldots,a_{n+1}=\sqrt{2+a_n}$ Prove (I've already proven $s_n$ is monotonically increasing and bounded above by 3) $\lim_{n \to ...
3
votes
2answers
106 views

The sum of the series $\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$ is an irrational number

Let $\{\epsilon_n\}$ be a sequence where $\epsilon_n$ is either $ 1$ or $-1$. How could I Show that the sum of the series $$\sum_{n=0}^{\infty}\frac{\epsilon_n}{n!}$$ is an irrational number.
11
votes
4answers
328 views

Prove the series $ \sum_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number

How can one prove that the series $\displaystyle \sum\limits_{n=1}^\infty \frac{1}{(n!)^2}$ converges to an irrational number? There's no need to use Taylor expansion, integrals or any ...
2
votes
2answers
170 views

The density — or otherwise — of $\{\{2^N\,\alpha\}:N\in\mathbb{N}\}$ for ALL irrational $\alpha$.

Problem Is there an irrational $\alpha\in\mathbb{R}\backslash\mathbb{Q}$ such that the set $S= \{\,\{2^N\alpha\} :N\,\in\mathbb{N}\}$ is not dense in $[0,1]$. Here $\{x\}=x-\lfloor x\rfloor$ is the ...
19
votes
3answers
530 views

Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational

Assume we have a sequence of rational numbers $a=(a_n)$. Assume we have a summation function $S: \mathscr {L}^1 \mapsto \mathbb R, \ \ S(a)=\sum a_n$ ($\mathscr {L}^1$ is the sequence space whose sums ...
4
votes
5answers
245 views

Will we get all real numbers if we add all limits?

Consider a set of all rational numbers from 0 to 1 inclusive. If we add to this set all limits of all convergent sequences of these numbers, will we obtain a set of all real numbers from 0 to 1?
5
votes
2answers
422 views

Irrational Numbers Containing Other Irrational Numbers

Does $ \sqrt{2} $ contain all the digits of $ \pi $ in order? Does it contain all the digits of $ \pi $ in order an infinite number of times? Does $ \pi $ contain all the digits of $ \sqrt{2} $ in ...
10
votes
2answers
1k views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in ...
0
votes
1answer
137 views

$a_n(x):=nx-\lfloor nx \rfloor$

i have $a_n(x):=nx-\lfloor nx \rfloor$ where $x$ is real. i want to show that if $x$ is rational, then $a_n(x)$ has finitely many cluster points, if $x$ is irrational, then every real $a$ with $0\leq ...
8
votes
1answer
234 views

Irrationality of a series

Here is a series in which $m \geq 2 $. I want to ask how to prove the below series is irrational: $$\sum _{n=1}^{\infty} \frac{1}{m^{n^2}}$$
4
votes
2answers
497 views

Use of the Reciprocal Fibonacci constant?

The Reciprocal Fibonacci constant ($\psi$) is defined as $$\psi=\sum_{k=1}^{\infty} \frac{1}{F_k}$$ where $F_{k}$ is the $k^{th}$ Fibonacci number. The irrationality of $\psi$ has been proven. ...
1
vote
0answers
116 views

Asymptotic behavior of $\sum_{j=1}^n \cos^p(\pi u j)$ for large $n$ and $p$?

Consider the sum $$S=\sum_{j=1}^n \cos^p(\pi u j),$$ where $n$ and $p$ are positive integers and $u$ is irrational. Let's say $p$ is even. I'm interested in the asymptotic behavior of this for $n$ ...
11
votes
1answer
136 views

Irrationality of Two Series

Show that if the integers $1<b_1<b_2<\cdots$ increase so rapidly that$$\frac{1}{b_{k+1}}+\frac{1}{b_{k+2}}+\cdots<\frac{1}{b_{k}-1}-\frac{1}{b_{k}},\quad k\geq 1,$$ then the number ...
14
votes
2answers
568 views

How come such different methods result in the same number, $e$?

I guess the proof of the identity $$ \sum_{n = 0}^{\infty} \frac{1}{n!} \equiv \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x $$ explains the connection between such different calculations. How ...
6
votes
3answers
708 views

Proving that a series converges to an irrational number

How do we show that if $g \geq 2$ is an integer, then the two series $$\sum\limits_{n=0}^{\infty} \frac{1}{g^{n^{2}}} \quad \ \text{and} \ \sum\limits_{n=0}^{\infty} \frac{1}{g^{n!}}$$ both converge ...
3
votes
2answers
398 views

Nature of the series: $ \sum\limits_{k=1}^{\infty} \frac{2^{n_k}}{(n_{k})!}$

Prove that if $\{n_k\}$ is a strictly increasing sequence of positive integers, then the sum of the series $$\sum_{k=1}^{\infty} \frac{2^{n_k}}{(n_{k})!}$$ is an irrational number. This is just a ...