Questions tagged [products]

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

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Computation of a finite product

I am preparing for job interviews, and in some old questions list I found this interesting one which I did not manage to crack yet: Compute the following products: $$ P_1 = \Pi_{0< i<j<\infty}...
Lenz's user avatar
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Characterizing congruences on the algebra of natural numbers

I'm trying to do Exercise 31 in Jan Rutten's book on coalgebras. The goal is to show that, given a characterization of congruences on the initial $N$-algebra $(\mathbb{N},[\text{zero},\text{succ}])$, ...
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Evaluating a summation of product

Show that for any integer $k>1$ $$ \sum_{\substack{i_j \in \{0, 1\} \forall j < k, i_k = 0}} \prod_{j = 1}^{k} \left(i_j + (-1)^{i_j} \frac{a+ (j - 1) c - c \sum_{\lambda = 0}^{j - 1} i_\lambda}{...
ParaN3xus's user avatar
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Show that among the numbers $x_{1}, x_{2} ,...,x_n $there are two whose product is at most equal to $- 1 / n$. [duplicate]

the question Let $n \in \mathbb{N}, n \geq 3,$ and $x_1, x_2,\dots,x_n \in \mathbb{R}$ such that $x_1+x_2+\dots+x_n = 0$ and $x_1^2+x_2^2+\dots+x_n^2 = 1$. Show that among the numbers $x_1, x_2,\dots,...
IONELA BUCIU's user avatar
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Computing the 21st derivative of $f(x) = \prod\limits_{n=1}^{5} \frac{x^n}{1-x^n}$ at $x=0$

Given a function, $f(x) = \prod\limits_{n=1}^5\frac{x^n}{1-x^n}$ Compute the value of 21st derivative of $f(x)$ at $x=0$ The answer given is $\frac{10}{21!}$. I proceeded with taking logarithm both ...
OpateItZOpatoOpate's user avatar
1 vote
3 answers
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Proving $\pi_1(X\times Y)=\pi_1(X)\times\pi_1(Y)$ by Categorical Methods

In J. P. May's A Concise Course in Algebraic Topology, Chapter 2, He states the following lemma is "immediate from the universal property of products": Lemma. For based spaces $X$ and $Y$, $\...
atzlt's user avatar
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Endomorphisms of a direct sum of abelian groups.

Let $I$ be a non-empty set and let $(A_i,+_i,0_i)$ for $i\in I$ be an abelian group, such that $Hom(A_i,A_j)=0$ for $i,j\in I, i\neq j$. Then $f$ is an endomorphism of a group $A=\bigoplus_{i\in I}{...
alpha's user avatar
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Why does a product of elements smaller than 1 converge to infinity?

I asked Wolfram Alpha to compute the following limit: $$ \lim_{n\to\infty} \prod_{i=1}^n (1- \frac{1}{n+2i}) $$ and the answer was $\infty$: I do not understand how this can be, as all elements in ...
Erel Segal-Halevi's user avatar
3 votes
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Finding Value of the Infinite Product $ \prod_{n=1}^{\infty}\frac{ e^\frac{1}{n}}{1+\frac{1}{n}}$

How I can find the value of the Infinite Product $$ \prod_{n=1}^{\infty}\frac{ e^\frac{1}{n}}{1+\frac{1}{n}}$$ I tried like this : $p_n =\prod_{k=1}^{n}\frac{ e^\frac{1}{k}}{1+\frac{1}{k}}= \prod_{k=1}...
A12345's user avatar
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Continuity and Product Spaces

I have the following basic question regarding continuous functions. Is the following statement correct? Fix nonempty toplogical spaces $A, B, C, D$ with $A \times C$ and $B \times D$ endowed with the ...
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Convoluted notation involving a double sum and a product $\sum_{m\geq0}\sum_{1\leq...<|x|;1\leq...<|y|}$ where $x$ and $y$ are sequences

Theorem 4 The discretized signature kernel over $k$, $$ \mathrm{k}^{+}: X^{+} \times X^{+} \rightarrow \mathbb{R}, \quad \mathrm{k}^{+}(x, y)=\left\langle\mathrm{S}^{+}\left(\mathrm{k}_x\right), \...
AnotherSherlock's user avatar
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1 answer
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Falling and rising factorial series identity

$$ \sum_{n=0}^\infty\prod_{m=1}^n\frac{x-m+1}{km} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{x+m-1}{(k+1)m} $$ I noticed this identity that relates the falling and rising factorials using a power series. ...
stackshifter's user avatar
6 votes
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Find all positive integers $n$ for which $\prod_{k=1}^n\left(k^4+\frac14\right)$ is the square of a rational number

Question Find all positive integers $n$ for which$$N=\left({1^4+\frac14}\right)\left({2^4+\frac14}\right)...\left({n^4+\frac14}\right)$$ is the square of a rational number. My attempt $$N=\prod_{k=1}^...
Mostafa's user avatar
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Simplifying tricky sum of products

Could someone please clearly write out how we get from this expression $$\log\left[\sum_{\mathbf Z}\left(\prod_{n=1}^N\prod_{m=1}^M\pi_m^{\mathbf{1}(z_n=m)}\mathcal{N}\left(\mathbf x_n;\mathbf{\mu}_m,\...
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Infinite products that are equal to their geometric product integral

Background In the following question and references therein, a number of "Sum equals integral" identities are described. For instance, we have $$ \sum_{n = -\infty}^{+\infty} {\rm sinc} (x)^{...
Max Muller's user avatar
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For a basis of product topology on the $X \times X \times \cdots$,could there be infinitely many sets between two sets that are not the $X$?

By definition, a basis of the product topology on $$ \prod_{\alpha \in J} X_\alpha $$ consists of sets of the form $$ \prod_{\alpha \in J} U_\alpha, $$ where the $U_\alpha \neq X_\alpha$ only ...
Pure 's user avatar
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Is the product of metric and semi-metric functions a semi metric?

I have a function - $B_{DGC}(x,y) = K_{(x,y)} \times e^{-(\frac{(I_x - I_y)^2}{2\sigma^2})}\times \frac{1}{d(x,y)} \times \frac{1}{\delta(x,y)_{DGC}}$. The components $K_{(x,y)}$ and $\frac{1}{\delta(...
Arijit De's user avatar
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Solving Kronecker product

I am trying to solve for X the following: $XA + B(X \otimes Y)Q=C$ is there a closed form solution to this? Thank you in advance. Note: How I can derive $B(X \otimes Y)Q$ with respect to X and then ...
Nnn A's user avatar
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1 answer
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Question regarding Transitivity of a Relation

Suppose we define a relation $R$ in the natural set $\mathbb N$ which says: $$(x,y)\in R\iff x^2-4xy+3y^2=0$$ and we would like to find which of the following properties does $R$ satisfy. My book ...
20DPCO190 Amanul Haque's user avatar
4 votes
1 answer
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Continuous analogue of the discrete simple continued fraction

Background The classical Riemann integral of a function $f : [a,b] \to \mathbb{R}$ can be defined by setting $$\int_{a}^{b} f(x) \ dx := \lim_{\Delta x \to 0} \sum f(x_{i}) \ \Delta x. $$ Here, the ...
Max Muller's user avatar
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Motivation and intuition behind the geometric product integral

According to the relevant wiki article, there are three types of product integrals. Type II, the geometric integral, is named as the continuous analogue of the discrete product operator. The geometric ...
Max Muller's user avatar
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Maximum product of numbers that add up to 10? [duplicate]

Let’s play a game where you have ten chips, and you want to split them up into stacks such that you want to maximise your score, which is the product of the size of each stack. What’s the best you can ...
Anon's user avatar
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Is there an Euler-Maclaurin-like formula for products?

Background The Euler-Maclaurin (E-M) formula is a formula for the difference between the sum and the integral of a real or complex continuous function on the interval $[m,n]$. It expresses this ...
Max Muller's user avatar
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1 vote
1 answer
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Understanding Why a Product of Cyclic Groups with Non-Coprime Orders is not Cyclic

Let $G_1, G_2, \ldots, G_t$ be finite cyclic groups, and define $G = G_1 \times G_2 \times \ldots \times G_t$. Let $n_j = |G_j|$, such that $|G| = \prod_{j=1}^{t} n_j$. For part (a) of the problem, we ...
user20194358's user avatar
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0 answers
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Inequality with Products and Sums

I need help to find a proof for the following inquality. Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that $$ \prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
Duns's user avatar
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21 votes
1 answer
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Every matrix is a product of two symmetric matrices

Let $\mathbb{F}$ be a field with char $\mathbb{F} \neq 2$. Let $A \in M_n(\mathbb{F})$. Does there exist symmetric matrices $B,C \in M_n(\mathbb{F})$ such that $A=BC$? The answer is yes when $\mathbb{...
MinaaaaaniM's user avatar
1 vote
1 answer
48 views

Prove $\frac{(n/2)!(n/2)!}{\left(\frac{n+i}2\right)!\left(\frac{n-i}2\right)!}=\prod_{j=1}^{i/2}\frac{\frac n2+1-j}{\frac n2+j}$

I have a textbook (Asymptopia by Joel Spencer, p.66) that states that $$\frac{(n/2)!(n/2)!}{\left(\frac{n+i}2\right)!\left(\frac{n-i}2\right)!} =\prod_{j=1}^{i/2}\frac{\frac n2+1-j}{\frac n2+j}.$$ The ...
YaGoi Root's user avatar
0 votes
2 answers
183 views

how to calculate $\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$

Question: how to calculate $$\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$$ My attempt Let $\arctan \theta =\frac{i}{2}\ln \left( \frac{i+\theta }{i-\theta } \right)$ $$S=\sum\limits_{k=1}^{...
Martin.s's user avatar
1 vote
1 answer
51 views

Sums and Products over sets of sets

Say I have a set of sets eg. $A:=\{\{1,2\},\{2,3\}\}$. Does this formula $$\sum_\limits{A_i \in A}\prod_{j\in A_i}x_j$$ yield $x_1x_2+x_2x_3$? Or does it even make sense to define a sum over a set of ...
beaver's user avatar
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-1 votes
1 answer
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Some pattern in the gap of the extrema/root of a product using Stirling approximation

Playing with Desmos calculator and Stirling approximation I define : $$x>0,f(x)=\prod_{n=1}^{M}\left(2-\frac{\Gamma(\frac{x}{n}+1)}{\sqrt{2\pi×\frac{x}{n}}\left(\frac{x}{e×n}\right)^{\frac{x}{n}}}\...
Miss and Mister cassoulet char's user avatar
5 votes
1 answer
235 views

How to find the coefficient of $x^k$ in the expression $\prod_{p=1}^n (x^p+1)^p$?

This Question asked on math over flow I tried to find the indefinite integral $$ f_n(x)=\int \prod_{k=1}^n \cos^k(kx)dx$$ by using Euler's formula and put $x=\frac{\ln y}{2i}$ I got $$ f_n(x)=-i2^{-\...
Faoler's user avatar
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3 votes
0 answers
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Techniques for proving this series identity?

$$e^{tx+h} = \sum_{n=0}^\infty\prod_{m=1}^n\frac{tx+h-\ln({\frac{t+1}{t}})(m-1)}{mt\ln({\frac{t+1}{t}})}$$ This is my attempt at parameterizing the function $ e^{tx+h} $ in terms of a power series ...
stackshifter's user avatar
0 votes
1 answer
111 views

Closed form for $(n^2+1^2)(n^2+2^2)...(n^2+(n-1)^2)$, $n\in\mathbb{N}$

I need closed form for the product $$P_n=(n^2+1^2)(n^2+2^2)...(n^2+(n-1)^2)$$ where $n\in\mathbb{N}$ I tried using the properties of Gamma function as follows: $$P_n= (n+i)(n-i)(n+2i)(n-2i)...(n+(n-1)...
Max's user avatar
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2 votes
1 answer
151 views

For $J=\{1,2,\dots,n \}$ is there an easy way to compute $\prod\limits_{i\in J | i \ne k} (k-i)$?

When I studied calculus at my university there is one question that I hated the most which is given a finite number of terms for some sequence find the $n-$th term. I hated this type of question ...
pie's user avatar
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2 votes
1 answer
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Solutions $X_{k}$ to the equation $\sum_{k=0}^N X_{k} (-2^k)^n = 1$

In the proof of the lemma in this paper, the author makes the following claim: Fix integer $N$ and let $n \leq N$. Then the solutions $X_0,\ldots,X_N$ to the equations $$\sum_{k=0}^N X_{k} (-2^k)^n = ...
Joseph Kwong's user avatar
1 vote
0 answers
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Defining a function, $\Pi(x_1, x_2, ... x_n)$ that will multiply all inputs together.

I want to define a function that outputs the product of all numbers I input. My reasoning is I have defined say 10 variables: $a$ through $k$. If I want to multiply them all together, writing: $a \...
melon's user avatar
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0 votes
2 answers
91 views

What is the product of the roots of a polynomial equation?

I'm trying to solve a question that says that for the equation $z^4 + pz^3 + 54z^2 - 108z + 80 = 0$, three of the roots of the equation are $3 + i, a$ and $a^2$. I know that for a quadratic equation $...
bang's user avatar
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2 votes
2 answers
81 views

Showing that the diagonal functor $\Delta:\mathbb{C} \to \mathbb{C} \times \mathbb{C}$ having a right adjoint implies $\mathbb{C}$ having products.

I started brushing up on my understanding of adjunctions and came across this well-known fact (rephrased in my own words): Let $\mathbb{C}$ be a category, and let $\Delta:\mathbb{C} \to \mathbb{C} \...
user11718766's user avatar
10 votes
2 answers
199 views

Cut a unit stick at $n-1$ random points. Expectation of product of fragment lengths is $\prod\limits_{k=n}^{2n-1}\frac1k$. Why?

On a straight stick of length $1$, choose $n-1$ independent uniformly random points. Cut the stick at those points, yielding $n$ fragments. Let $\mathbb{E}_n$ be the expectation of the product of ...
Dan's user avatar
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1 vote
3 answers
168 views

Show $8 \sin \frac{4 \pi}{9} \sin \frac{2 \pi}{9} \sin \frac{\pi}{9} = \sqrt{3}$.

Please provide guidance on how to solve the product to sum question. I have also attached my attempt which was unsuccessful... Show $8 \sin \frac{4 \pi}{9} \sin \frac{2 \pi}{9} \sin \frac{\pi}{9} = \...
racer234's user avatar
0 votes
0 answers
28 views

Help proving the product of 1+ positive real numbers is greater than its sum [duplicate]

I am trying to prove the following inequality: Given real numbers $a_i\geq0, i=1,2,\ldots n$, then $\prod^n_{i=1}(1+a_i)\geq \sum^n_{i=1} a_i$. I've tried to do it using induction but I wasn't able to ...
frysauce's user avatar
11 votes
1 answer
204 views

$e$ is hidden in Pascal's (binomial) triangle. What is hidden in the trinomial triangle, in the same way?

In Pascal's triangle, denote $S_n=\prod\limits_{k=0}^n\binom{n}{k}$. It can be shown that $$\lim_{n\to\infty}\frac{S_{n-1}S_{n+1}}{{S_n}^2}=e$$ What is the analogous result for the trinomial triangle? ...
Dan's user avatar
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A hypothesis For $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$

(Hypothesis) For integer $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$, then the following identity, whether or not to be established: $$ \prod_{j=1}^m \text{Li}_{l_j}\left[ x_j \right] = \sum_{k=0}^...
user avatar
14 votes
1 answer
246 views

Showing that, in Pascal's triangle, the product of the numbers along each median is always the same.

On Pascal's triangle with any number of rows, draw the three medians. For each median, calculate the product of the numbers that zig-zag along that median. The three products are always equal! (proof ...
Dan's user avatar
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3 votes
0 answers
72 views

Closed form of product of sedenions

I'm a math student and I'm taking an algebra course. The professor introduced us to the field of quaternions ($\mathbb{Q}$), I became very curious about the topic and I saw that in addition to ...
Efesto's user avatar
  • 1
18 votes
2 answers
893 views

Trying to calculate a limit with a finite product and WolframAlpha disagrees with my logic.

I want to calculate $$\lim_{N \to \infty} \prod_{j=1}^{N} \frac{2j}{N + j + 1} \,.$$ Here is my logic: For the range of the product operator ($1 \leq j \leq N $) the inequality $2j < N + j + 1$ is ...
Jose's user avatar
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2 votes
0 answers
131 views

Covariance of the product of two random variables with another random variable

Let $X\sim \mathrm{Bin}(p_x,1)$ , $Y\sim \mathrm{Bin}(p_x,1)$ and $V\sim \mathrm{Bin}(p_v,1)$ be three binomial random variables, that are NOT independent from each other. Note that $X$ and $Y$ are ...
CafféSospeso's user avatar
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0 answers
53 views

Multiplicative Harmonic Series

Is there any approximation formula for this equation? I have been trying to find approximations for it so I can create an approximation formula myself for something else, but for some reason I can't ...
Monke's user avatar
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2 votes
1 answer
126 views

Let $A_{k}=\{0,... ,n\}\setminus\{k\}.$ How to prove $\sum_{k=0}^{n}\left[(-1)^{k+1}\prod_{\substack{i,j\in A_{k}\\i<j}}(a_{i}-a_{j})\right]=0$?

Let $A_{k}=\{0,1,\ldots,n\}\setminus\{k\}$ for each $k=0,1,\ldots ,n$. I think that the following equality is true for all $n\in\mathbb{N}, n\geq 2$ : \begin{align} \sum_{k=0}^{n}\left[(-1)^{k+1}\...
Musube o's user avatar
-1 votes
1 answer
86 views

What does x equal? [closed]

I have 4 numbers -- a, b, c, and d -- that I can pair up in 6 different ways to multiply -- ab, ac, ad, bc, bd, cd .when I do, I get products in the set {5, 6, 10, 12, 18, x}. what does x equal ? My ...
Moaz Nasem's user avatar

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