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

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0
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3answers
110 views

What is the dot product of two or three vectors graphically or visually?

I don't understand what the dot product actually is. I understand when and where to use it, but when it comes to proving things with it, I don't really grasp what it actually is making it difficult ...
1
vote
2answers
222 views

What is the mistake in this proof of product rule of differentiation?

I was trying to derive the product rule of differentiation which states: If $y=u\cdot v$, then, $y'=u'\cdot v+v'\cdot u$. So I assumed it like this: $y=u+u+u+\cdots$ ($v$ number of terms of $u$) ...
4
votes
1answer
218 views

example diagram of pullbacks and fiber products

I am going through Category Theory for Scientists. I am on section 2.5.1 Pullbacks. I am having trouble visualizing a pullback. Earlier in the book the author gives a nice diagram of an example of ...
0
votes
1answer
52 views

Summation in 104 Number Theory problems

There's a paragraph of 104 Number Theory problems, on page $9$ that says: From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$ using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
2
votes
1answer
144 views

A polynomial equality problem

$a_1,a_2,a_3,\ldots,a_n,a_{n+1}$ are fixed real numbers in $(-1,\infty)$. $x_1$ and $x_2$ are fixed real numbers in $(0,1)$. Is it possible to prove that there exists or doesn't exists a real number ...
3
votes
3answers
163 views

How find this value $\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$

Find the value $$\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$$ I know this :How find this $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^6}\right)$ and maybe can find the $2k+1$? can you someone konw ...
0
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2answers
78 views

How to prove a product of k consecutive integers is always a multiple of k?

How to explain and prove that a product of k consecutive integers is always a multiple of k!.
2
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0answers
26 views

Fourier Series from product of to functions

I have to calculate the Fourier Series of $x\sin(x)$ beeing $2\pi$ periodic on $[-\pi,\pi]$and i did it the standard way. But then i wanted to solve the problem with multiplication of two fourier ...
0
votes
3answers
41 views

Dot Product and vector length

Hi! I am working on some online homework for my calc2 class that covers the dot product and I am really struggling with this one question. I understood how to solve part a, because we covered that ...
0
votes
1answer
20 views

Condition for the product of the ratio of the elements of two sequences to be greater than 1.

I have the following product: $$\prod_{n=1}^N \frac{A_n}{B_n}$$ , where $A_n$ and $B_n$ are the nth element of the finite sequences {$A_x$} and {$B_x$} respectively. I'd like to know the conditions ...
1
vote
1answer
49 views

Prove $F_n(z)=\frac1{2i}\left(\left(1+\frac{iz}n\right)^n-\left(1-\frac{iz}n\right)^n\right)…$

In my textbook there is a proof for the following If $n=2m+1$ with $m\in\mathbb N$, then we can write ...
2
votes
0answers
31 views

Is there a sort of “two-sided semidirect product”? [duplicate]

Let $G,H$ be groups. Suppose we have both an action of $G$ on $H$, and an action of $H$ on $G$, both non-trivial. Let "$\cdot$" define the former action, and $\circ$ define the latter. What can we ...
0
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0answers
31 views

How do I do the math necessary to make these five matrices multiplied together equal the result shown?

I'm currently studying the math involved with rotating vertices around an arbitrary axis in 3D space. For this, I have found the following page to be very helpful: ...
2
votes
0answers
199 views

The logarithm of a product

Let $p$ be a prime number, $C\in \mathbb{N}$ and C is not a square. Then define $$F=\prod_{|z| \leq \sqrt{\frac{x}{2}} \atop |y|\leq \sqrt{\frac{x}{2D}}}{|z^2-Cy^2|}.$$ Note that we omit the term with ...
1
vote
3answers
100 views

Product of “reversed” numbers

Consider any 2 binary numbers, e.g.: 10101011 ; 11111101 and their product, say P. "Reverse" (mirror image) all the digits of the 2 numbers, e.g.: ...
4
votes
2answers
142 views

Showing $\sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64}$

I would like to show that $$ \sin{\frac{\pi}{13}} \cdot \sin{\frac{2\pi}{13}} \cdot \sin{\frac{3\pi}{13}} \cdots \sin{\frac{6\pi}{13}} = \frac{\sqrt{13}}{64} $$ I've been working on this for a few ...
1
vote
1answer
57 views

Expanding a product formally.

Let $a_1,...,a_n$ be real numbers. I don't know how to formally expand the following product $$ \prod_{k=1}^n(1+a_k) $$ I'm guessing something like (edited) $$1+\huge\sum_{k=1}^n \; ...
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
81 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
71 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 ...
7
votes
3answers
370 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
56 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
115 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
23 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
287 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
65 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 ...
3
votes
1answer
175 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 ...
1
vote
1answer
110 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
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2answers
79 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
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 $n$?
0
votes
1answer
38 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
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 ...
5
votes
3answers
145 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
77 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
74 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
52 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
44 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
72 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
69 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
430 views

Can Pi prod be expressed using Sigma Notation?

My question is simple (but difficult for me): $\prod(x)$ be expressed interms of $\sum (x)$
2
votes
2answers
761 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
votes
3answers
81 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
47 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
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 ...
0
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2answers
194 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?
2
votes
2answers
80 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
151 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
44 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.
7
votes
4answers
138 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 ...