5
votes
1answer
187 views

Exponentials of rational numbers

Does there exist an $$0<x<1$$ such that $$\forall q \in \mathbb{Q^+}$$ $$q^x \in \mathbb{Q^+}$$
0
votes
1answer
24 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
43 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
-1
votes
0answers
26 views

The best generalization of a sequence [closed]

I have an equation, I will build sequences from this equation and I define a generalized equation and look for the best generalization of the sequences ! I will present an example and ask you if it is ...
0
votes
0answers
31 views

Zero to power Zero (Zero ^ Zero) indeterminable or not? [duplicate]

I want to know Zero power to Zero equal to 1 or Indeterminable. I think it cannot be exist. Please explain with proper mathematical definitions.
2
votes
1answer
67 views

Simultaneous rational approximation of real numbers in (0,1)

I have a simple question the rational approximation of real vectors. Dirichlet's simultaneous approximation theorem states: Given any $d$ real numbers $\alpha_1,\ldots,\alpha_d$ and a natural ...
5
votes
2answers
54 views

Non analytic numbers

We know that some real numbers (actually, most of them) are not algebraic and the proof of this fact is beautiful: algebraic numbers, like polynomials with integer coefficients, are countable, ...
0
votes
1answer
39 views

How to show that a real number has a finite decimal representation?

How to show that a real number has a finite decimal representation (one that ends with an infinite sequence of zeros) if and only if it can be represented as a rational number m/n where n has no prime ...
2
votes
0answers
49 views

Expected error due to the tablemakers' dilemma

[note: to me, this does not seem like a question for m.se, but on mathoverflow it has been retroactively closed, with very little indication of why or what might be corrected... and waiting for ...
0
votes
1answer
34 views

Convergence of a modified sum of prime reciprocals for all $s \in \mathbb{C}$?

It is known that $\displaystyle \sum^\infty_{p \in \mathbb{P}} \frac{1}{p^s}$, with $\mathbb{P}$ the set of primes, only converges for $\Re(s) > 1$. The following sum of primes seems to converge ...
0
votes
0answers
27 views

Real numbers correspond bijectively to decimal expansions

Prove that positive real numbers correspond bijectively to decimal expansions no terminating in an infinite string of 9's as follows. The decimal expansion of $x \in R$ is $N.x_1x_2...$, where $N$ is ...
4
votes
3answers
90 views

limit of rational sequence

Let $a$ and $b$ be two real number. Assume that there exits two real sequences $a_n, b_n$ such that $$ \lim a_n=1, \lim b_n=1$$ and all $$\frac{a- a_n}{b-b_n} $$are rational numbers. Is it true ...
0
votes
0answers
50 views

linear diophantine equation and limit

Let $a_1, a_2, \dots, a_l$ are natural numbers and $\text{gcd}(a_1, a_2, \dots, a_l)=1$. Let $A_n$ is a number of non-negative integer solutions of $a_1x_1+a_2x_2+\dots+a_lx_l=n$. Prove that $\lim ...
3
votes
1answer
124 views

More identities of the Ramanujan Double factorial type.

Ramanujan discovered the following identity $$x=\sum_{n=0}^\infty (-1)^n ...
4
votes
1answer
106 views

Estimating integrals involving $\pi(x)$

While solving an exercise in analytic number theory, I ran into difficulty of estimating an integral of the form $\displaystyle\int_{1}^{x} \frac{\pi(t)}{t} dt$ where $\pi(x)$ is the prime counting ...
1
vote
1answer
37 views

Measure of a set??

I looking to find a measure of the following set: $A=\{ (a_1,a_2,...a_k) \in \mathbb{R}^k : |1+a_1z_1+a_2z_2+...+a_kz_k|<\delta \}$ and where $z_i \in \mathbb{Z}$. I believe the measure of this set ...
3
votes
5answers
149 views

Are transcendental numbers dense in $\mathbb R$? What is algebraic number? Is $cos(\pi/13)$ algebraic?

Are transcendental numbers dense in $\mathbb R$? What is algebraic number? Is $\cos(\pi/13)$ algebraic?
1
vote
0answers
49 views

Cardinality of sum-set

Let $X\subset R$ and $Y \subset R$ where X and Y have finite cardinalities. Let also, $a,b \in R$. How to show that $|aX+bY|=|X||Y|$ almost everywhere (measure of $(a,b) \subset R^2$ such that ...
0
votes
1answer
56 views

General polynomial form of a factorial?

Is anyone aware if there is a general polynomial form of a factorial? For instance; $2!=2^2-2^1$ $3!=3^3-3(3^2)+2(3^1)$ $4!=4^4-6(4^3)+11(4^2)-6(4^1)$ $5!=5^5-10(5^4)+35(5^3)-50(5^2)+24(5^1)$ ...
5
votes
1answer
109 views

express rational number as sum of squares of unit fraction

Let $q$ be a rational number with $0\lt q\leq\dfrac{\pi^2}6-1$. Then show that there exists a set $S\subset \{2,3,4,\dotsc\}$, such that $$q=\sum_{n\in S}\frac1{n^2}$$ I have no clue about it. Could ...
2
votes
2answers
170 views

Can We Represent Every Real Number Using Only Finite Memory?

This question arises from a comment I recently read in another question. My question is whether we can represent every real number using only finite memory. I will clarify what I mean by represent ...
2
votes
1answer
105 views

Different proofs of $\lim_{x\to \infty}\left(1+ \frac{1}{n}\right)^n =e$

I recently was teaching my friend about the number $e$. I introduced him the number by using the compound interest thing . Then I wrote down the general result -$$\lim_{x\to \infty}\left(1+ ...
25
votes
1answer
490 views

Show $(1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\cdots)^2 = 1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49} + \cdots$

Last month I was calculating $\displaystyle \int_0^\infty \frac{1}{1+x^4}\, dx$ when I stumbled on the surprising identity: $$\sum_{n=0}^\infty (-1)^n\left(\frac{1}{4n+1} +\frac{1}{4n+3}\right) = ...
2
votes
1answer
73 views

Irrational Natural Density

Are there any sets of natural numbers with irrational natural density? I.e., does there exist a set $A\subset \mathbb{N}$ such that $$ \lim_{n->\infty} \frac{|A \cap [1,n]|}{n} \not\in ...
6
votes
1answer
44 views

Can You Construct a Syndetic Set with an Undefined Density?

Let $A \subset \mathbb{N}$. Enumerate $A = \{A_1, A_2,...\}$ such that $A_1 \le A_2 \le ...$. We say that $A$ is syndetic if there exists some $M \geq 0$ such that $A_{i+1} - A_i \le M$ for all $i ...
3
votes
1answer
95 views

Diophantine Equation Question

Find all ordered pairs (x,y) of rational numbers x, y such that the equations $2x^5 = x^2y^4 + 9y^5$ and $6y^3 = 3x^3 + xy^3$ hold simultaneously. My try: Multiplying the second equation by xy, we ...
1
vote
1answer
42 views

Prove that for $n\ge1$, $\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},…,a_2,a_1\rangle\right)^{-1}$

Prove that for $n\ge1$, $$\xi-\frac{h_n}{k_n}=(-1)^nk_n^{-2}\left(\xi_{n+1}+\langle 0,a_n,a_{n-1},...,a_2,a_1\rangle\right)^{-1}$$ In addition, show that ...
3
votes
3answers
106 views

What does the integer span of one irrational, and one (possibly irrational) real number look like in $\mathbb{R}$?

My title was rejected a few times, here is what it was initially: If you take two real numbers- one irrational and one possibly irrational - how close does their $\mathbb{Z}$ span come to any ...
2
votes
1answer
61 views

Cosine and sine dense in unit circle

We may assume the following theorem: Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$. Assume $\lambda$ is ...
2
votes
1answer
47 views

Density with irrational number and trig function

We may assume the following theorem: Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$. Consider the points ...
5
votes
2answers
120 views

If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?

This appears on problem 1 of chapter 1 in Stein & Shakarchi's Real Analysis: Given an irrational $x$, one can show (using the pigeon-hole principle, for example) that there are infinitely many ...
4
votes
1answer
111 views

Generalization of an inequality $0\lt e^6-{\pi}^4-{\pi}^5\lt 0.00002$

Question : Is the following true? For any $n\in\mathbb N$, there exists a triple $(k,l,m)\ (k,l,m\in\mathbb N)$ such that $$0\lt e^k-{\pi}^l-{\pi}^m\lt{10}^{-n}.$$ Motivation : A friend ...
2
votes
2answers
91 views

Formula for number of $m^{th}$ power sums less than $x$

Let the class of numbers that is calculated by the Faulhaber's formula as-$$S_m(n)=\sum\limits_{i=1}^{n}i^m$$ For $m=1$ it gives the triangular numbers and so on. I want to define a function ...
2
votes
2answers
109 views

Do Hyperreal numbers include infinitesimals?

According to definition of Hyperreal numbers The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + 1 ...
0
votes
2answers
94 views

Using induction to prove that every integer can be written in a particular form

(a) Use induction to prove that every integer $n$ can be written in the form: $$n = \beta_0 3^0 + \beta_1 3^1 + \cdots + \beta_{r-1} 3^{r-1} + \beta_r 3^r$$ where $r$ is a non-negative ...
0
votes
1answer
68 views

Proof or disproof of this integral in a series.

I need a proof or disproof of- $$\frac{\pi^{8}}{3150}=\sum_{k=0}^{\infty} \int_{0}^{\infty}\frac{t^{3} e^{-4(k + 1)t}}{1-e^{-4(k + 1)t}}\,dt$$
1
vote
1answer
70 views

An upper bound for $-\frac{\zeta'}{\zeta}(s)-\frac{1}{s-1}$

Let $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$. We have $\frac{\zeta'}{\zeta}(s) = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^s}$ for $s>1$, where $\Lambda$ stands for the von Mangoldt function ...
2
votes
5answers
149 views

What's so special about $e$? [duplicate]

If someone with not much mathematics in his luggage asks me: What is so special about $\pi$? then off course I have an answer. Even if $i$ would be the subject (I allready see him gazing at my ...
0
votes
1answer
82 views

Supremum and Infimum of $\,S=\{(-1)^k + 2^-k| k≠0 \,\,\text{and}\,\, k>1\}$

Consider $S=\left\{((-1)^k + 2^{-k}|k\in\mathbb{N}k≠0\right\}$ Determine the $Sup(S)$ and $Inf(S)$ and justify. So far I have that that: $-1 \lt (-1)^m + 2^{-m}$ $\forall m=2k+1$ (The odd powers of ...
0
votes
0answers
105 views

Zeros of a power series

Suppose we have a power series with (real or complex) coefficients $\sum_{n \geq 0} a_n x^n$ (that has nonzero radius of convergence). Can one say something about its zeros in terms of the ...
2
votes
0answers
103 views

Integer values of the Riemann function - II

For what value of $n \ge 2$ can we have an real $x > 0$ such that both the numbers $$ \zeta\Big(1+\frac{1}{x}\Big) \text{ and } \zeta\Big(1+\frac{1}{nx}\Big) $$ are positive integers.
4
votes
4answers
198 views

How calculate $\pi$ to an accuracy of 10 decimal places?

Let $a=3.00000000001234...$ (irrational number) If $\overline{a}=3.00000000001$ (approximation $11$ places) then $|a-\overline{a}|<10^{-11}$ Note that the reciprocal is not satisfied: If ...
11
votes
1answer
214 views

How to prove that $\sqrt{2}+\sqrt{3}>\pi$

Does someone know a other proofs (using properties of $\pi$) of following inequality: $$\sqrt{2}+\sqrt{3}>\pi$$ First proof: the area of ​​regular 48-gon circumscribed to the unit circle is ...
1
vote
1answer
87 views

Dense subsets of $R$

I have the following problem: Let $(a_n)$ and $(b_n)$ two sequences of natural numbers such that, $a_n\to +\infty$ and $b_n\to +\infty$. Prove that: $$K=\{\pm\frac {a_n}{b_m}:n,m\in\mathbb{N}\}$$ ...
4
votes
1answer
142 views

Prime harmonic series

We have following identity: ($p$ is a prime number) $$\left(1+\frac{1}{p}\right)\sum_{k=0}^n\frac{1}{p^{2k}}=\sum_{k=0}^{2n+1}\frac{1}{p^k}$$ Now, How to derive the following inequality from the above ...
2
votes
2answers
147 views

Showing limit of a sequence $0, \dfrac12, \dfrac14, \dfrac38, \dfrac5{16}, \dfrac{11}{32}, \dfrac{21}{64},…$

How do you show the convergence of the following 2 sequences? $0, \dfrac12, \dfrac14, \dfrac38, \dfrac5{16}, \dfrac{11}{32}, \dfrac{21}{64},...$ and $1, \dfrac12, \dfrac34, \dfrac58, ...
4
votes
4answers
1k views

Supremum and Infimum of this set..

Am I correct in taking the first few values for $m$ and $n$ as $1,2$ then $2,3$ then $3,4$ respectively? How do I go about finding the supremum and infimum for this sequence? Also how do I ...
3
votes
1answer
71 views

Bertrand's postulate proof

Regarding http://michaelnielsen.org/polymath1/index.php?title=Bertrand%27s_postulate I think the last inequality should be $4^{n/3}\le(2n+1)(2n)^{\sqrt{2n}}$. But even when the RHS is decreased from ...
2
votes
1answer
65 views

A sequence with near constant auto-correlation?

Suppose $$ x[n]= \begin{cases} x_n &, n \in P\\ 0 &, n \notin P \end{cases} $$ where $P \subset \{0,1, \cdots,N-1 \}$ and $|P|=K$ and $x_n \geq 0$. for this sequence these equations hold: $$ ...
4
votes
4answers
169 views

Can anyone give me a counterexample to this statement? [duplicate]

Statement: Let $n$ and $m$ be two irrational numbers. Then $n^m$ is always irrational. I think this statement is correct, otherwise can someone give me a counterexample? Thanks!