# Tagged Questions

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### Bernoulli number analog identity

I am doing work on Bernoulli numbers. Let $$b(x)=\sum_{k=0}^\infty{\frac{B_k}{k!}x^k}=\frac{x}{e^x-1}$$ From this, by taking derivatives of both sides we get the identity $$b(x)^2=(1-x)b(x)=xb'(x)$$ ...
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### How is the set of all closed intervals countable?

I am trying to figure out the answer to the problem: Show that the set of all closed intervals $[a,b]$ with $a,b \in \mathbb{Q}$ is countable. Now I know that the interval $[0,1)$ for example is ...
Let $(x_{n})_{n\ge 1}$ be an increasing sequence of positive integers and $\displaystyle{\overline{x}_{n}:=\dfrac{1}{n}\sum_{i=1}^{n}x_{i}}$. Suppose furthermore that $\forall\varepsilon\gt 0, \ \ ... 2answers 134 views ### Can$\sum_1^n 1/k$be arranged so that it is an integer for infinitely many$n?$It's well known that when$n>1:$$$\sum_{k=1}^n \frac{1}{k}\not\in \mathbb{N}$$ But if we are allowed to rearrange the series, we can for instance can get: ... 0answers 18 views ### Identifying or bounding the zeros of the composition of two generating functions Given two generating functions $$G(a_n;x)=\sum_{n=0}^\infty a_nx^n \quad\text{ and }\quad H(b_n;x)=\sum_{n=0}^\infty b_nx^n,$$ what techniques are available for locating, or finding bounds on, the ... 1answer 95 views ### Computing infinite product over primes How can I compute $$\prod_p \left(1+\frac{k}{p}\right)\exp(-k/p)$$ where$0<k<e$and the product is over all primes$p$? Background L. G. Sathe proved [1] that there are $$... 2answers 332 views ### A series with only rational terms for \ln \ln 2 We all know that$$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$Do you know a series with only rational terms for$$\ln \ln 2 = ?$$Let's exclude base expansions with non ... 0answers 40 views ### Approximating real vectors in the unit hypercube by rational vectors in the unit hypercube I am currently looking for an error bound for an problem where I have to approximate a real vector in the unit hypercube using a rational vector in the unit hypercube. Specifically: given a ... 0answers 50 views ### a special function for count Let be f:(\mathbb{N} \setminus \left\{0,1 \right\})^2 \rightarrow \mathbb{N} function that f(a,k)=\text{total numbers of }n \in \mathbb{N} \text{ that } \frac{a^n}{n^k} \le 1 . My question is: ... 0answers 78 views ### Points in a general Cantor set We often look at the Cantor set with the construction that keeps removing the middle thirds of the remaining line segments at each iteration. Corresponding to this construction, we can determine ... 4answers 515 views ### Geometric intuition for \pi /4 = 1 - 1/3 + 1/5 - \cdots? Following reading this great post : What are some interesting cases of \pi appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis ... 1answer 113 views ### Can any real number be expressed as an integral combination of e and \pi? Prove or disprove: For any real number x, there exist integers a and b such that ae + b\pi=x. It certainly seems improbable, but how does one prove it? 1answer 39 views ### Does sequences related to function for lcm(1,2,3 \cdots n) exists? This just came out of curiosity let$$L(n)=lcm(1,2,3 \cdots n)$$and I know that we can write this with the help of some product involving primes and all . But what I am interested is in Does ... 3answers 69 views ### What is meant by a function that varies in only one direction? My professor mentioned in class today about a "function that varies in only one direction". To clarify, he said he meant a function which is a function of a linear combination of the variables. That ... 1answer 82 views ### Asymptotic of a sum evaluation as x \to \infty Let be the sum$$ \sum_{n\le x}[x/n]=g(x) $$where [x] means floor function. My best try for asymptotic is g(x) \sim x\log (x)+\gamma x +1 where I have used the asymptotic [x] \sim x ... 1answer 187 views ### The closed form of \sum_{n=1}^{\infty} \left(\frac{1}{\lfloor\sqrt{3n}\rfloor^2}-\frac{1}{3n}\right) I need some ideas to exploit for finding the closed form of$$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor\sqrt{3n}\rfloor^2}-\frac{1}{3n}\right)$$0answers 48 views ### Rational approximation bound for real numbers in (0,1) I am working on a problem that related to rational approximation of real numbers. I am looking for a bound of the form: Given a positive real number, \alpha \in (0,1), there exist positive ... 0answers 154 views ### Conjecture on OEIS A167055 OEIS A167055 Numbers n such that 12n + 5 is prime. 0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,... are items of OEIS A167055. I conjecture that the set of the sum of every two items of this ... 3answers 212 views ### Does \sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p} converges? Does$$\sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p}$$converges ? I know that the following \sum_{p\in\mathbb P} \frac{1}{p} diverges, we can find proofs on Wikepedia Divergence of the ... 3answers 202 views ### If \sum\frac1{a_n} is convergent, then irrational? \{a_n\} is a strictly increasing sequence of positive integers such that$$\lim_{n\to\infty}\frac{a_{n+1}}{ a_n}=1$$If \sum\limits_{n=1}^\infty\frac1{a_n} is convergent, can one conclude ... 1answer 67 views ### The function f(t)=2+\sin(t)+\sin(t\sqrt2) The function f defined on \mathbb{R} by$$f(t)=2+\sin(t)+\sin(t\sqrt2)$$can never reach 0. Can we find some sequence (t_n)_{n\geq0} such that$$\lim_{n \to \infty}f(t_n)=0 \ \ \ ?$$Or in ... 2answers 84 views ### Non uniform continuity of a function and almost periodicity We say that a continuous function f:\mathbb{R}\to\mathbb{C} is almost periodic in the sense of Bohr if: For every sequence (t'_n)_{n\geq0}, there's a sub-sequence (t_n)_{n\geq0} such that ... 1answer 32 views ### About the implicit funtion in a holomorphic situation. Let f(x,y) be a polonomial with integral coefficients which has a zero (a,b)\in \mathbb{R}^2 such that the partial derivative respect to y at this point is nonzero. Then by the implicit function ... 1answer 368 views ### How to find \int_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx I am trying to calculate the integral$$ I_n=\int \limits_0^\infty \prod_{k=1}^n \frac{\sin \frac{x}{2k-1}}{\frac{x}{2k-1}}\mathrm dx. $$(I have literature on this, if people want). Note, we can ... 2answers 95 views ### Looking for references I am looking for reference on the following problem. Let S= \{ ax_1+bx_2 \mid x_1 \in X_1 , x_2 \in X_2, \} where X_1,X_2\subseteq \mathbb{R}^n and a,b \in \mathbb{R}. Note that a and b ... 1answer 200 views ### Exponentials of rational numbers Does there exist an$$0<x<1$$such that$$\forall q \in \mathbb{Q^+}q^x \in \mathbb{Q^+}$$1answer 26 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 ... 3answers 47 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 ... 0answers 39 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. 1answer 117 views ### Existence of a 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 for every ... 2answers 60 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, ... 1answer 49 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 ... 0answers 51 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 ... 1answer 38 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 ... 0answers 95 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 ... 3answers 93 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} $$are rational numbers. Is it true ... 0answers 60 views ### linear diophantine equation and limit Let a_1, a_2, \dots, a_l are natural numbers and \text{gcd}(a_1, a_2, \dots, a_l)=1. Let A_n is a number of non-negative integer solutions of a_1x_1+a_2x_2+\dots+a_lx_l=n. Prove that \lim ... 1answer 186 views ### More identities of the Ramanujan Double factorial type. Ramanujan discovered the following identity$$x=\sum_{n=0}^\infty (-1)^n ... 1answer 131 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 ... 1answer 40 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 ... 5answers 229 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? 0answers 61 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 ... 1answer 72 views ### General polynomial form of a factorial? Is anyone aware if there is a general polynomial form of a factorial? For instance;$2!=2^2-2^13!=3^3-3(3^2)+2(3^1)4!=4^4-6(4^3)+11(4^2)-6(4^1)5!=5^5-10(5^4)+35(5^3)-50(5^2)+24(5^1)$... 1answer 124 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 ... 2answers 205 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 ... 1answer 121 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 -$$\lim_{x\to \infty}\left(1+ ... 1answer 528 views ### Show (1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\cdots)^2 = 1+\frac{1}{9}+\frac{1}{25}+\frac{1}{49} + \cdots Last month I was calculating \displaystyle \int_0^\infty \frac{1}{1+x^4}\, dx when I stumbled on the surprising identity:$$\sum_{n=0}^\infty (-1)^n\left(\frac{1}{4n+1} +\frac{1}{4n+3}\right) = ... 1answer 81 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 ... 1answer 58 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 ...
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 ...