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

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1answer
93 views

Probability distribution of the product of random numbers

For applied mathematics to evolutionary biology I am often faced to have to describe a probability distribution function (PDF) which results from the product of a function in which a parameter is ...
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2answers
75 views

Is there any number $n$ such that $nm=0$, $n\neq 0$, and $m\neq 0$?

I answered a question about whether zero is prime or composite on Khan Academy a while ago. Since then, two people have commented on my answer, asking another question that I don't know the answer to. ...
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2answers
75 views

Can we get the formula for $\prod\limits_{k=0}^n{(1+2^k)^2}$ in terms of $n$?

Can we get the formula for $\prod\limits_{k=0}^n{(1+2^k)^2}$ in terms of $n$?
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2answers
222 views

Proof: Categorical Product = Topological Product

Is there a nice way to prove that the categorical product equals the topological product? What I mean is the following: Starting with a given family of topological spaces $X_i$ and any topological ...
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1answer
37 views

Restrictions on a Matrix-Vector product

Suppose I have a $m\times n$ matrix $\mathbf M$, and a unit vector $\hat v$, of dimension $n$. What restrictions do I need to apply to $\mathbf M$ so that $\lVert \mathbf M\cdot \hat v\lVert \leq 1$ ...
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2answers
60 views

Product of $1-\operatorname{cis}(2k\pi/n)$

I'm in a question about polygonals and got stuck at a part. I have to prove that $$\prod_{k=1}^{n-1} \left(1 - \operatorname{cis}(\frac{2k\pi}{n})\right) = n$$ I've tried to multiply it to make ...
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3answers
143 views

Expressing $\prod_{k=1}^n \left( k - \frac{1}{2} \right)$ using the gamma function

I want to express $$\prod_{k=1}^n \left( k - \frac{1}{2} \right)$$ using the gamma function. I think this is equivalent to $\left(k-\frac{1}{2}\right)!$ so I set $a=k-1$ and then used the identity ...
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1answer
70 views

Dot product of taylor series $\sqrt{1+x}$

I have to prove that $$ \sum_{k=1}^n \alpha_k \cdot \alpha_{n-k+1} = 0, $$ where $n>2$ and $\alpha_k$ is the k-th member in taylor series of $\sqrt{1+x}$. Namely, $$ \alpha_k = ...
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2answers
71 views

Lagrange polynomials sum to one

I've been stuck on this problem for a few weeks now. Any help? Prove: $\sum_{i=1}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}=1$ The sum of lagrange polynomials should be one, otherwise affine ...
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1answer
51 views

Understanding relation between Product and Summation Notation

So I am given the following: $n = \sum_{i=1}^{k}m_{i}$ I am also given $x = \sum_{i=1}^{k}log(m_{i}) = log\prod_{i=1}^{k}m_{i}$ I was only given the first part, however I believe that is a ...
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1answer
43 views

Replace $n$ sets with two sets (set theoretic equality)

Let $A_0,\dots,A_{n-1}$ be sets for some whole $n>0$. Take $A'_{0, i} = A_i$ and $A'_{1, i} = \bigcup ( \{ A_0, \ldots A_{n - 1} \} \setminus \{A_i\})$ for $i=0,\dots,n-1$. Prove (or disprove) $$ ...
1
vote
1answer
39 views

Norm of a Matrix-vector product

Suppose I have vector $\vec x \in \mathbb R^n$ and matrix $\mathbf M$ of dimension $m\times n$. Is there an alternative expression for $\lVert \mathbf M \cdot \vec x \lVert$ that includes $\lVert \vec ...
0
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2answers
68 views

Proving $\prod_{i=1}^n (\frac{1}{i} + 1) = n+1$

Prove using a direct proof that $$\prod_{i=1}^n \left(\frac{1}{i} + 1\right) = n+1$$ Okay, so I think I have done it correctly using an inductive proof: Base case: $(1+\frac11)=2$, ...
3
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1answer
349 views

Can Pi prod be expressed using Sigma Notation?

My question is simple (but difficult for me): $\prod(x)$ be expressed interms of $\sum (x)$
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2answers
599 views

How to interchange a sum and a product?

I have this expression: $$\sum_{\{\vec{S}\}}\prod_{i=1}^{N}e^{\beta HS_{i}}=\prod_{i=1}^{N}\sum_{S_{i}\in\{-1,1\}}e^{\beta HS_{i}} \qquad (1)$$ Where $\sum_{\{\vec{S}\}}$ means a sum over all possible ...
2
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3answers
80 views

Prove that $(1-\frac{1}{2^2}\cdots 1-\frac{1}{9\,999^2})(1-\frac{1}{10\,000^2})=0.500\,05$ [duplicate]

Prove that $\displaystyle\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{9\,999^2}\right)\left(1-\frac{1}{10\,000^2}\right)=0.500\,05$ Here are all my attempts to ...
2
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2answers
46 views

If $a\neq 1$, find $(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)$.

If $a\neq 1$, find $$(a+1)(a^2+1)(a^4+1)\ldots(a^{2^n}+1)$$ Or i.e. If $a\neq 1$, find $\prod_{i=0}^n(a^{2^i}+1)$. It really does seem like ...
2
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1answer
31 views

Union of Cartesian squares

I am trying to find sufficient and necessary conditions for a relation to be representable as a union of Cartesian squares: $\bigcup_{i\in I}(X_i\times X_i)$ for some family $X_i$ of sets. One ...
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2answers
128 views

swap summation and multiple

In which case can we swap summation and multiple? ie. $$\sum_{i=1}^n\prod_{j=1}^na_{ij}=\prod_{j=1}^n\sum_{i=1}^na_{ij}$$ if we can't swap like this, please tell me how can we swap them?
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0answers
110 views

Way to split up product of summation

If I have $\sum_{n=1}^{\infty}f(x)g(x)$, is there any way to split this up? Thanks.
2
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2answers
79 views

When can we write the square of a matrix as the product of the matrix and its transpose?

I often see something like $(A - B)^2$ being written as $(A - B)(A - B)^T$ . Here $A$ and $B$ are two matrices. I can see that this is possible when $A$ and $B$ are scalars (i.e) single element ...
3
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1answer
128 views

Direct product commutes with direct sum?

Do direct products commute with the direct sums of vector spaces? Basically is $\underset{i \in I}{\prod} \underset{j \in J}{\bigoplus}M_{i,j} \cong \underset{j \in J}{\bigoplus}\underset{i \in ...
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2answers
27 views

Derivative: Which rule to use first?

$f(x)=x^7(5+8x)^3$ Would I go about finding the derivative of this problem by using the chain rule first, and then the product rule? Or would I do the opposite? Step by step instructions would be ...
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1answer
41 views

Roots of unity product

For each $n \in \mathbb N, n \geq 3$ calculate the product of all the n roots of unity. Or to say it in a more stric way: $$\prod_{w \in G_n^*}w$$ Being $G_n^*$ the primitive roots of the unity.
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4answers
136 views

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to?

What does $\prod_{n\geq2}\frac{n^4-1}{n^4+1}$ converge to? As far as I can tell, this has no closed-form solution (not saying much, I don't know much math), but a friend of mine swears he saw a ...
1
vote
1answer
670 views

Average of products VS. product of averages

I have a problem at work where prices of things are determined by multiplying together a series of factors. For example assume each price is made up of three factors, $A, B, C$, so that $\text{Price} ...
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0answers
178 views

Product of standard normal and uniform random variable

I'm trying to find the PDF of the product of two random variables by first finding the CDF. I don't know where I'm going wrong. Let $X\sim N(0,1)$ and $Y\sim Uniform\{-1,1\}$ and let $Z = XY$, then: ...
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1answer
545 views

Prove that the product of some numbers between perfect squares is $2k^2$

Here's a question I've recently come up with: Prove that for every natural $x$, we can find arbitrary number of integers in the interval $[x^2,(x+1)^2]$ so that their product is in the form of ...
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1answer
30 views

Proof $\prod_{i = 1}^n \frac{n + i}{2i-3} = 2^n(1-2n)$ using inducction

i'm trying to solve this, using induction. The base step is easy, there's no difficult there. The problem comes in the inductive step, I got to demonstrate that: $$\prod_{i = 1}^{n+1} \frac{n+ 1 + ...
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0answers
33 views

any approximation for $\sum_{i_3=3}^{n-k+3}\sum_{i_4=i_3+1}^{n-k+4}\sum_{i_5=i_4+1}^{n-k+5}\cdots\sum_{i_k=i_{k-1}+1}^{n} 1, (n \gg k)$?

Is there any approximation for $$\sum\limits_{i_3=3}^{n-k+3}\sum\limits_{i_4=i_3+1}^{n-k+4}\sum\limits_{i_5=i_4+1}^{n-k+5}\cdots\sum\limits_{i_k=i_{k-1}+1}^{n} 1, \quad (n \gg k)$$ ? We know that ...
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5answers
80 views

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$

Is there any good approximation for $\prod_{i=3}^k (n-i)$? $(n \gg k)$ I know that $\prod_{i=3}^k (n-i) < \prod_{i=3}^k n = n^{k-2}$ Also a tighter upper bound is appreciated.
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2answers
75 views

Basic Cartesian prodcuts

I am having some issues grasping basic ideas of Cartesian products. I am reading a PDF my professor gave us explain sets/Cartesian products. If $\mathbb{R}\times \mathbb{R}$ can be written as ...
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3answers
265 views

Vector multiplication. Difference between scaler and dot product?

We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the ...
0
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1answer
24 views

Let $a_i$ , $1\le i\le n$ be non-negative real numbers. Let S denote their sum.Pick out the true statements:

Let $a_i$ , $1\le i\le n$ be non-negative real numbers. Let S denote their sum.Pick out the true statements: (a)$\prod_{k=1}^{n}{(1+a_k)\ge1+S}$ ...
0
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1answer
28 views

Exponential equivalent for geometric space

I'm just starting a foray into geometric algebra and calculus so that I can develop a geometric version of the standard arithmetic neural net. Specifically when calculating the error function for a ...
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2answers
114 views

Coefficients of $(x-1)(x-2)\cdots(x-k)$

I'm interested in the coefficients of $x$ in the expansion of, $$ (x-1)(x-2)\cdots(x-k) = x^k + P_1(k) x^{k-1} + P_2(k)x^{k-2} + \cdots + P_k(k),$$ Where $k$ is an integer. In particular I am ...
0
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2answers
56 views

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate the limit

Let $a > 0$ and let $k \in \mathbb{N}$. Evaluate: $$\lim_{n\to \infty}a^{-nk}\prod ^k_{j=1}\left(a+\frac{j}{n}\right)^n$$ Clueless on this problem. Seek your help.
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2answers
68 views

What are the properties around Pi products

There are some well-known evaluations around summations like $\sum 1$ or $\sum i^2$ but what are these properties for products, specifically something like $\prod_{i=0}^{n-1} (n-i)$. I basically have ...
3
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0answers
104 views

Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
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2answers
81 views

Asymptotics of $1-\prod\limits_{i=0}^{k-1}\left(1-\frac{i}{2^k}\right)$

I am interested in the asymptotics of $$1-\prod_{i=0}^{k-1}\left(1-\frac{i}{2^k}\right).$$ As a rough piece of mostly incorrect work this looks a little like $$1-\prod_{i=0}^{k-1}e^{-i/2^k} = ...
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3answers
77 views

Finding limit of a product.

Prove:$$\lim_{n \to\infty }\frac{1}{n}\left[\prod_{i=1}^{n}(n+i) \right ]^{\frac{1}{n}}=\frac{4}{e}$$ I tried using Squeeze Theorem but can't go beyond $1<L<2$. $$\lim_{n\to\infty} \left( 1 + ...
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2answers
111 views

Forumula for calculating $1\cdot 2\cdot3\cdot4\cdot5\cdot\ldots\cdot n$

I'm looking for a formula to calculate the (product?) of an arithmetic series. Something like this: $$\frac{n(a_1+a_n)}{2}$$ which is used to get the sum of the series, expect instead of all the ...
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1answer
34 views

Boundary of product cartesian

What's the boundary of $\Omega\times (a,b)$, where $\Omega$ is an open bounded subset of $\mathbb R^n$ ?
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2answers
115 views

The floor of a product of fractions

Evaluate: $ \displaystyle \Bigg \lfloor \prod_{n=0}^{248} \frac{33+8n}{29+8n} \Bigg \rfloor= \Bigg \lfloor \frac{33}{29} \times \frac{41}{37} \times \frac{49}{45} \times\ ...\ \times ...
2
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0answers
83 views

Multiplicative group into ring operation

My question is simple, though it proves to be much more difficult than it sounds. Suppose I want to find a binary operation to add extra structure to a multiplicative group (so it becomes a ring). ...
5
votes
5answers
103 views

Showing that $\lim_{n\to\infty}\frac{1\cdot 3\cdot 5\cdots (2n-1)}{(2n)^n} = 0$

Ok, so I want to show that $$\lim_{n\to\infty}\frac{1\cdot 3\cdot 5\cdots (2n-1)}{(2n)^n} = 0.$$ Here is what I have tried so far: \begin{align} \notag \lim_{n\to\infty}\frac{1\cdot 3\cdot 5\cdot ...
3
votes
4answers
52 views

Sum and product of integers conserving order

I have the feeling this is true, but can't prove it: $$\sum_n^An\lt\sum_n^Bn\implies\prod_n^An\lt\prod_n^Bn$$ Where $A\subset\mathbb N-\{0, 1\},B\subset\mathbb N-\{0, 1\}$ Example: ...
0
votes
1answer
50 views

What's the maths symbol for alternation of product and sums?

Is there a mathematics symbol for referring to the equation below? (((((((((((100*y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x = 0, x = 9.8 I've tried using capital ...
1
vote
4answers
111 views

Why is this equal to 1?

Why is $$ \prod_{i=4}^0 (4i -1) = 1 $$ At least according to: http://www.wolframalpha.com/input/?=prod_{i%3D4}^0+%284*i+-+1%29 It is rather unintuitive, why would the product even be defined? One ...
0
votes
2answers
38 views

How to repesent n x m multiplication into symbol notation?

I am not a mathematician and so I might not be using the right terms. I have a vector of n components and another vector of m components ...