Tagged Questions

For questions about the evaluation of finite products, or their properties. For infinite ones, use "infinite-products" tag.

2answers
321 views

1answer
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Box topology is finer than the uniform topology on $\mathbb{R}^\mathbb{N}$

This time, I wish to show that the box topology is finer than the uniform topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ However, the problem here is that ...
1answer
27 views

Uniform topology is finer than the product topology on $\mathbb{R}^\mathbb{N}$

I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ We know both spaces are metrizable: The ...
2answers
50 views

2answers
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Inequality and Induction: $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$ [duplicate]

I needed to prove that $\prod_{i=1}^n\frac{2i-1}{2i}$ $<$ $\frac{1}{\sqrt{2n+1}}$, $\forall n \geq 1$ . I've atempted by induction. I proved the case for $n=1$ and assumed it holds ...
7answers
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Induction and convergence of an inequality: $\frac{1\cdot3\cdot5\cdots(2n-1)}{2\cdot4\cdot6\cdots(2n)}\leq \frac{1}{\sqrt{2n+1}}$

Problem statement: Prove that $\frac{1*3*5*...*(2n-1)}{2*4*6*...(2n)}\leq \frac{1}{\sqrt{2n+1}}$ and that there exists a limit when $n \to \infty$. , $n\in \mathbb{N}$ My progress LHS is ...
3answers
4k views

Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots ...
2answers
62 views

About “The product of the six numbers surrounding any interior number in Pascal’s triangle is a perfect square”

The current Futility Closet has this statement: "The product of the six numbers surrounding any interior number in Pascal’s triangle is a perfect square." Here is the link with a nice illustration: ...
2answers
41 views

Relation between HCF, LCM and product of multiple numbers [duplicate]

It is well known that for two numbers $a$ and $b$, $$\text {lcm} (a,b)\times \text {hcf} (a,b)=ab$$ Does there exist a similar equality/ inequality between HCF, LCM and product of multiple numbers? (...
0answers
41 views

Multiplication of polynomials of the same degree

Consider polynomials of the form $$p(x)=x^{n-2r}\sum_{i=0}^ra_ix^{2i},$$ where \begin{align} r&=n/2, \quad n \quad \text{even},\\ r&=(n-1)/2, \quad n \quad \text{...
7answers
284 views

Why does $(-1) \times (-1)$ give +1? [duplicate]

Why is $(-1) \times (-1)=+1$ ? What is the intuitive concept ? My second question : How can I show that no triangular number can be of the form $3n-1$ ?
1answer
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Is this matrix going to be real or complex?

I hope that this is the right forum where to post this question (and not here). I have a Chi-Square Kernel Matrix (using the second version, which is positive-definite) ...
1answer
38 views

Is it possible to calculate the Average of Products from the Product of Averages?

I have two sets of data - one $X$ measuring the unavailability at a site, the other $Y$ measuring the number of antennas at each site. I want to calculate the overall average unavailability as the ...
1answer
45 views

Finding the product of a prime function…

If we take the primes $p_k < n$, and raise them to the highest power possible such that $(p_k)^{r_k} \le n$, what is the lower bounds on $\prod{ (p_k)^{r_k} }$? In other words, what are the ...
10answers
8k views

Is $0! = 1$ because there is only one way to do nothing?

The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or ...
1answer
24 views

Composition method and constructing a relation.

Let $R = \{(1, 5), (2, 2), (3, 4), (5, 2)\}$, $S = \{(2, 4), (3, 4), (3, 1), (5, 5)\}$, and $T = \{(1, 4), (3, 5), (4, 1)\}$. Find (1)$\quad R ∘ S$ (2)$\quad T ∘T.$ (3) $\quad T∘S$ ...
1answer
61 views

Value of finite product based on empty set

How does one evaluate the following product if the set S happens to be empty? \begin{aligned} f(n)= n \prod_{x \in S} \left(1-\frac{1}{x}\right) \end{aligned} Is the value simply n or is it ...
1answer
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1answer
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Understanding components of a vector

I learned that we can get the component of a vector in any direction using the dot product. The problem I have is the meaning of the term component itself. The component of a vector $\vec A$ in the ...
1answer
27 views

Is there anything known in general about upper and lower bounds for $\prod_{i\leq n\vee p_n>k}(p_i-k)$

I have no specific reason to ask this question other than seeing that it comes up quite often when I'm playing around with prime numbers. Let $$f(n,k)=\prod_{i \leq n\vee p_n>k}(p_i-k)$$ Where $p_i$...
1answer
34 views

Source of faulty reasoning in expectation of product of random variables?

For iid $\xi_i>0$, with $\mathbb E[\xi_i]=1$, what is $\mathbb E[\prod_i^M\xi_i]$? Approach 1: $\mathbb E[\prod_i^M\xi_i]=\prod_i^M\mathbb E[\xi_i]=1$. There is another approach for $M\gg1$ with ...
1answer
672 views

Product of Fibonacci numbers

I'm looking for the asymptotic approximation of the product of the first $n$ Fibonacci numbers. Does there exist a tight approximation for these kind of things?
1answer
63 views

The limit of consecutive positive integers which are the product of n primes.

The maximum length of a string of consecutive primes is 2: that is, the primes 2, 3. This is easily proven, as no even number other than 2 is prime. In contrast, consider the set of numbers which ...
1answer
53 views

Product as the sum of powers times a symmetric polynomial: What's the name of this property and what is it used for?

I noticed that the product of a group of positive integers $N$ with $n$ elements can be expressed as the sum of powers of the smallest member of $N$ times some (what I later found out be called) ...
1answer
48 views

Proving ${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)$

Let $A=2^{2^{-x}}$ and $B=2^{2^{-x}+1}(1+2^{2^{-1}})(1+2^{2^{-2}})\cdots(1+2^{2^{-x+1}})$ Showing $x\ge2$ $${\pi\over 2}=2\tan^{-1}\left({1\over A}\right)+\tan^{-1}\left({1\over B}\right)\tag1$$ ...
1answer
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If two infinite sets have the finite closed topology, then their product does not have the finite closed topology.

Let $X_1$ and $X_2$ be infinite sets and $T_1$ and $T_2$ be the finite-closed topology on $X_1$ and $X_2$, respectively. Show that the product topology, $T$, on $X_2 \times X_2$ is not the finite ...
1answer
44 views

Proof of $\sin nx=2^{n-1}\prod_{k=0}^{n-1} \sin\left( x + \frac{k\pi}{n} \right)$

I have seen this identity on Wolfram mathworld and in a comment to another similar trigonometric proof: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I can't seem to find a ...
1answer
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Modified Sum of Products

A given number k is to be expressed as a sum of products of integers keeping in mind that the integers used in above process do not exceed their cumulative sum as 100. For e.g., k = 19 can be ...
1answer
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Summation is to integration, as the large product operator is to…? [duplicate]

The integral is defined many ways but one that I am aware of is the Riemann Integral(?) which looks sorta like $\sum^n_{i=0} f(a +i\frac {b-a}n)*\frac {b-a} n$. An interesting thought is "is there a ...
1answer
22 views

Product of all Square Roots, taken only Decimal Digits

How and where could I compute the decimal reminder of a product of square roots times ten: $$Dr\left( \prod_{x=1}^{k}x^\frac{1}{2} \right) \times 10$$ Where $k$ is a power of $10$. I would like to ...
1answer
56 views

How to prove that $e^{-\gamma}=\prod_{n=1}^\infty\left(1+\frac1n\right)e^{-1/n}$

Suppose we defined the Gamma function $$\frac1{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^\infty\left(1+\frac zn\right)e^{-z/n}$$ where $\gamma$ is just a constant. I want to prove that $\Gamma(1)=1$, so I ...
0answers
67 views

Is this a finite product describing the partial harmonic series sums?

http://mathworld.wolfram.com/EulerProduct.html In the second last line, it gives a product P(n). Is this supposed to be describing the finite terms of the harmonic series sum? I don't see how it does....
1answer
42 views