Tagged Questions

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

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0
votes
2answers
36 views

Product approximation

In this biology textbok I found the following approximation: $$\prod_{i=1}^{k-1}1-\frac{i}{2N} ≈ 1-\frac{{k \choose 2}}{2N} $$ Can you help me to understand this approximation and help me to ...
1
vote
1answer
42 views

Chain or product rule for heat diffusion equation

A portion of the heat diffusion equation for a 1-D solid is given as: $$\frac{1}{r} \frac{\partial}{\partial r} \left(r \; k \frac{\partial T}{\partial r} \right)$$ Apparently this can be expanded ...
2
votes
2answers
69 views

Closed form for $\prod_{k=1}^n \binom{k^2+2k}{k^2}$

Does anybody know how I can find a closed form for the expression $$ \prod_{k=1}^n \binom{k^2+2k}{k^2}? $$ There are many ways to handle such things, but with sum instead of product. Here, I have no ...
6
votes
3answers
249 views

How to find the value of $\sqrt{1\sqrt{2\sqrt{3 \cdots}}}$?

I thought up this question recently, and I think I've figured out the partial sum: $$ S_n := \left(n\prod_{k=2}^{n-1} k^{2^{n-k}}\right)^{2^{-k}}. $$ But I don't even quite know if I'm on the right ...
2
votes
1answer
49 views

Finding numbers whose product is a particular number?

Is there a standard way to formulate and evaluate the following? Basically, I want to find 100 possibly distinct (some numbers can be repeated) real numbers ($0 < 1 + \frac{n_i}{100} < 5$) such ...
0
votes
1answer
61 views

Multiplying Sigmas(sums)

I would be grateful if someone please rewrite or expand this please. I have problem multiplying two sigmas ($\sum $) $$ (d(n)-\sum_{k=-\infty}^{\infty} h_k x(n-k)) \times ...
0
votes
1answer
20 views

Probability and Production equation translation

I know that the pi is like a summation except multiplication instead of addition and that P(x) means the probability of, but I'm having trouble putting it all together, esp the $w_i$ such that $w_1, ...
2
votes
1answer
85 views

Product of $n$ i.i.d. random variables

Let the variable $Z$ equal $Z = XY$ where $X$ and $X$ are two i.i.d. continuous random variables which distributions are given by $f_X()$ and $f_Y$. The distribution of $Z$ is given by: $$f_Z(z) = ...
2
votes
2answers
50 views

Matrix-Vector Product

Suppose I have the expression $\lVert\mathbf B \cdot\hat n\lVert=1$, where $\mathbf B$ is a matrix and $\hat n$ is a unit vector (both can have any dimensions, as long as they are compatible). What ...
0
votes
0answers
67 views

Nested sums and products

I am trying to determine how to express this series in a general form as a summation of products: \begin{equation} \begin{aligned} (p_{i1}p_{j1})(1-p_{i2}p_{j2})(1-p_{i3}p_{j3}) \mbox{ } &+ \\ ...
2
votes
1answer
96 views

How can I generate the products of two three-digit numbers in descending order?

While experimenting with different solutions to a little programming exercise, I generated an array with the products of all two three-digit numbers (i.e. 100 to 999). Since I wanted to process those ...
0
votes
1answer
79 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 ...
0
votes
2answers
69 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. ...
1
vote
2answers
74 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 $n$?
1
vote
2answers
166 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 ...
0
votes
1answer
30 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 $\lVert \mathbf M\cdot \hat v\lVert \leq 1$ ...
3
votes
2answers
53 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 ...
5
votes
3answers
126 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 ...
0
votes
1answer
60 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 = ...
0
votes
2answers
52 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 ...
0
votes
1answer
41 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 ...
0
votes
1answer
41 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
31 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
votes
2answers
61 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
votes
1answer
162 views

Can Pi prod be expressed using Sigma Notation?

My question is simple (but difficult for me): $\prod(x)$ be expressed interms of $\sum (x)$
0
votes
0answers
33 views

How to derive this inequality containing power series? (equations are contained in the body) (Changed)

As I read a paper, I don't know how do I derive inequality, $$\frac{\prod^N_{i=1}4^{b_i/N}(1-4^{-b_i})^{1/N}}{12}\ge \frac{4^{R/N}}{16} \\ \frac{\prod^N_{i=1}(4^{b_i}-1)^{1/N}}{12}\ge ...
1
vote
2answers
272 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 ...
0
votes
0answers
16 views

Distribution of Matrix and Vector products

Given the following expression: $$ \vec w = (\mathbf M\cdot\vec u) + (\vec v\cdot\vec u) $$ Where $\mathbf M$ is a matrix of dimension $n\times m$, $\vec v$ and $\vec u$ are vectors of dimension ...
2
votes
3answers
73 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
votes
2answers
45 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
votes
1answer
29 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 ...
0
votes
2answers
69 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?
0
votes
0answers
50 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
votes
2answers
75 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
votes
1answer
85 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 ...
0
votes
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 ...
0
votes
1answer
38 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.
6
votes
4answers
123 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
353 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} ...
0
votes
0answers
111 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: ...
15
votes
1answer
529 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 ...
1
vote
1answer
29 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 + ...
1
vote
0answers
32 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 ...
2
votes
5answers
79 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.
0
votes
0answers
38 views

Can we convert a sum-of-products into strictly sums?

So I start with a sum-of-products: $$f = \sum_{k=a}^b{\prod_{j=c}^d{g(j,k)}}$$ I'm wondering if we can somehow convert this into a sum of sums, ie: $$f = \sum_{k=a}^b{\sum_{j=c}^d{h(j,k)}}$$ It's ...
1
vote
2answers
66 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 ...
2
votes
3answers
129 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
votes
1answer
23 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
votes
1answer
21 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 ...
4
votes
2answers
105 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 ...