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Interesting infinite product $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$

I have found an interesting family of infinite products. The most interesting one of them being: $\sqrt{2}-1=\dfrac{1\cdot7\cdot9\cdot15\cdot17\cdot23\cdots}{3\cdot5\cdot11\cdot13\cdot19\cdot21\cdots}$...

sequences-and-series infinite-product
 asked by Lithium Score of 21
 answered by Martin R Score of 4

In the following, I'll present a curious double integral that despite its daunting look has a very nice closed form, $$\int _0^{\pi/2}\int _0^{\pi/2}\cot (x) \csc ^2(y) \log (\cos (y)) \log \left(1-2 \... real-analysis calculus integration sequences-and-series definite-integrals  asked by user97357329 Score of 14  answered by Zacky Score of 3 Strapping down a cylinder I want to strap down a big heavy cylinder on a flatbed truck. The strap is attached to the truck bed as shown in the picture and also behind. Will the strap slip off as in the next picture? PS. This ... geometry  asked by cdupont Score of 14  answered by Intelligenti pauca Score of 15 How does the common definition of a function satisfy the precise definition of a function? Before my reading on Linear Algebra by Hoffman/Kunze, I was under the impression that a function was "a rule (or mathematical object) that maps/assigns each x \in X (domain) to an element y \... functions definition  asked by IDK Score of 12  answered by Ethan Bolker Score of 5 Is a function \mathbb{R}^n \to \mathbb{R} which has a closed and connected graph necessarily continuous? It has been proved here and here that a function \mathbb{R} \to \mathbb{R} which has a closed and connected graph is continuous. This fact is also proved in a nice article by Burgess. I don't know ... general-topology  asked by Geoffrey Sangston Score of 11 Applications of Category Theory in Abstract Algebra Almost every text on Category theory uses categories such as Ab, Grp, and so on as examples to work with but can category theoretic methods actually help us understand the structures better? In ... abstract-algebra category-theory  asked by Acharyachakit Score of 11  answered by Qiaochu Yuan Score of 11 f(xy) = f(x+y) : A variation of an old problem I have seen the following problem in various places, such as Toppr and Quora. Suppose that the function f:\mathbb{N}\to\mathbb{N} satisfies f(x+y)=f(xy) for all x,y\in\mathbb{N}. Show that f ... contest-math recreational-mathematics functional-equations  asked by Timothy Chow Score of 10  answered by Mastrem Score of 2 Greatest hits from previous weeks: Does \pi contain all possible number combinations? \pi Pi Pi is an infinite, nonrepeating (sic) decimal - meaning that every possible number combination exists somewhere in pi. Converted into ASCII text, somewhere in that infinite string of ... number-theory irrational-numbers pi  asked by Chani Score of 774 How many even numbers of four digits can be formed with the digits 0,1,2,3,4,5 and 6 no digit being used more? My attempt to solve this problem is: First digit cannot be zero, so the number of choices only 6 (1,2,3,4,5,6) The last digit can be pick from 0,2,4,6, so the number of choices only 4 Second ... combinatorics  asked by akusaja Score of 4  answered by akusaja Score of 3 Find the coordinates of a point on a circle I have a circle like so r, with angle \theta to the y-axis"> Given a rotation θ and a radius r, how do I find the coordinate (x,y)? Keep in mind, this rotation could be anywhere between 0 and ... geometry trigonometry circles rotations  asked by CoderTheTyler Score of 41  answered by André Nicolas Score of 48 Probability of winning a prize in a raffle My work is having it's annual Christmas raffle today. 1600 tickets have been sold, and there are 40 prizes to win. I have bought ten tickets. What are the odds I will win a prize? While an initial ... probability combinatorics binomial-coefficients recreational-mathematics problem-solving  asked by Clarkey Score of 11  answered by Srivatsan Score of 22 What does the dot product of two vectors represent? I know how to calculate the dot product of two vectors alright. However, it is not clear to me what, exactly, does the dot product represent. The product of two numbers, 2 and 3, we say that it ... geometry vectors  asked by Saturn Score of 137  answered by King Squirrel Score of 93 Probability of getting exactly 2 heads in 3 coins tossed with order not important? I have been thinking of this problem for the post 3-4 hours, I have come up with this problem it is not a home work exercise Let's say I have 3 coins and I toss them, Here order is not important ... probability  asked by Max Score of 7  answered by Sammy Black Score of 10 If a club has 24 members, In how many ways can 4 officers be chosen from the members of the club? I understand the concept of combinations and permutations. However, I am not getting how to apply the formulas. I believe understanding exactly how to do this would help.A club has 24 members. a. In ... permutations combinations  asked by XxTIBZxX Score of 4  answered by DeepSea Score of 7 Can you answer these questions? Notation related question about a two variable functor fixing one of its variables. The following question is taken from "Arrows, Structures and Functors the categorical imperative" by Arbib and Manes \color{Green}{Background:} \textbf{(1)} \textbf{Definition:} A ... category-theory notation functors  asked by Seth Score of 1 Show that a(\gcd (n,k)) is generated from roots of generating function of \sum _{h=0}^{\infty } \left(\sum _{k=1}^n x^{h n+k} a(\gcd (n,k))\right) Let a(n) be the Dirichlet inverse of the Euler totient function:$$a(n) = \sum\limits_{d|n} d \cdot \mu(d) \tag{1}$$And let the matrix T be:$$T(n,k)=a(\gcd(n,k)) \tag{2} Compute the ordinary ...

number-theory polynomials roots generating-functions gcd-and-lcm
 asked by Mats Granvik Score of 1

Bound the sum of the prime factors for a special case

For an integer $n = \prod_{i=1}^k p_i$ where all $p_i$'s are distinct primes and $k \ge 2$, can we give an upper bound of the sum of these prime factors, namely $\sum_{i=1}^k p_i$ ? (or some ...

elementary-number-theory prime-numbers combinatorial-number-theory
 asked by hx1a Score of 1
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