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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2answers
61 views

How to show that the determinant of this matrix is in a nice product factorization,

Show that $$det \begin{bmatrix} 1 & 1 & \cdots &1 \\ \lambda_1 & \lambda_2 & \cdots &\lambda_n \\ \lambda^2_1 & \lambda^2_2 & \cdots ...
8
votes
1answer
301 views

Why is this sum equal to $0$?

While solving a differential equation problem involving power series, I stumbled upon a sum (below) that seemed to be always equal to $0$, for any positive integer $s$. $$ \sum_{k=0}^s \left( \frac{ ...
2
votes
1answer
24 views

Does this vector product, based on indexing with a powerset, have a name?

Given two vectors $\vec{u}$, $\vec{v}$ indexed by $2^X$ for some finite set $X$, define $\vec{u} \star \vec{v}$ as the vector of similar type whose dimension indexed by $S \subseteq X$ is: ...
0
votes
2answers
27 views

Result of product with n=0

It's a simple question but I couldn't find informations about it and I'm starting to learn product sequences. I noticed (using WolframAlpha) that: $\prod_{i=x}^{0}{f(i)}|_{x > 0} = 1$ Why is ...
1
vote
0answers
11 views

How can I prove the following inequality?

Let be $N_{n+1}(x)=\prod_{i=0}^n(x-x_i)$. Now I have to prove that $$||N_{n+1}(x)||_{\infty,[-5,5]}\leq n!\frac{h^{n+1}}{4},\qquad h:=\frac{5-(-5)}{n}=\frac{10}{n}.$$ I've started with ...
1
vote
1answer
50 views

Let $f(x)=1/x$ and prove that $f[x_0,x_1,…,x_n]=\prod_{i=0}^nx_i^{-1}$. [closed]

Let $f(x)=1/x$ and prove that $f[x_0,x_1,...,x_n]=\prod_{i=0}^nx_i^{-1}$. I'm sure how to approach this or even how/why we need $f(x)=1/x$. Any solutions or hints are greatly appreciated.
3
votes
1answer
32 views

Prove that the product $\prod_{i=1}^n \left(1-(1/2)^i\right)\ge\left (1/4 + 1/2^{n+1}\right)$ for any integer $n$

Expanded out this would be $(1-(1/2))(1-(1/4))(1-(1/8)) \cdots (1-(1/2^{n})) \ge (1/4 + 1/2^{n+1})$. I'm currently working to solve this problem but I cannot come to a reasonable conclusion. I am ...
3
votes
0answers
41 views

$n$ integers, $a_i a_j +1$ all perfect squares [closed]

Find all numbers $n$ so that there exists $n$ integers $a_1, a_2, ..., a_n$: $a_i \ge 2$ and $a_i\cdot a_j +1 (\forall i\not = j)$ are all perfect squares.
1
vote
1answer
148 views

Product of numbers is even , when an unbiased die rolled?

An unbiased die is thrown $n$ times. The probability that the product of numbers would be even is $1/(2n)$ $1/[(6n)!]$ $1−6^{−n}$ $6^{−n}$ None of the above. My attempt : we have $3$ even ...
0
votes
1answer
34 views

Undoing Capital Pi

I have an equation of which I'm trying to undo capital pi notation to simply get the expression. The equation is $c=\prod_{i=1}^k n-(i-1)$ How would I undo this?
0
votes
3answers
47 views

How to Differentiate $x^7(7x+5)^6$

I am trying to differentiate $f(x) = x^7(7x+5)^6$. So far I have done the following steps: 1) Use the product rule, which is $(x^7(6(7x+5)^5))+((7x^6)(7x+5)^6)$ 2) Factor out $x^6$ and $(7x+5)^5$ ...
1
vote
1answer
32 views

Show that n $\sum_{i=1}^n c_i$ and $\prod_{i=1}^n c_i$ are rational numbers.

Let $n$ be a positive integer and let $A ∈ M_{n×n}(\mathbb{Q})$. Let $c_1,...,c_n$ be the list of (not necessarily distinct) eigenvalues of $A$, considered as a matrix in $M_{n×n}(\mathbb{C})$. Show ...
0
votes
1answer
36 views

Solving without induction

$$\prod_{k=1}^n\cos\frac{x}{2^k}=\frac{\sin{x}}{2^n\sin\frac{x}{2^n}}$$ I tried to prove this without induction, but I can't come up with any idea. My teacher solved it with induction, which is the ...
0
votes
2answers
56 views

6 different positive numbers; sum = product

As the title says, I want to know if there exists a set of 6 DIFFERENT POSITIVE numbers such that their sum equals their product. (a+b+c+d+e+f=abcdef)
3
votes
1answer
72 views

Find the value of the infinite product $(3)^{\frac{1}{3}} (9)^{\frac{1}{9}} (27)^{\frac{1}{27}}$…

I'm not sure if this is meant to be a contradiction but if a product is an infinite product it does not mean that the value if infinity? Or is the word infinite product just misleading. I let: $ ...
3
votes
4answers
98 views

Express the following product as a single fraction: $(1+\frac{1}{3})(1+\frac{1}{9})(1+\frac{1}{81})\cdots$

I'm having difficulty with this problem: What i did was: I rewrote the $1$ as $\frac{3}{3}$ here is what i rewrote the whole product as: ...
1
vote
0answers
28 views

Algebraic proof that $\det AB = \det A \cdot \det B$ using Leibniz formula for determinants [duplicate]

The Leibniz formula for determinants allows us to express an $n \times n$ matrix determinant as a sum over permutations in $S_n$: $$\det A = \sum_{\sigma \in S_n} \operatorname{sgn}(\sigma) \cdot ...
3
votes
0answers
111 views

Asymptotic solutions of a sparsely perturbed recurrence relation

Recurrence relation I am trying to find approximate solutions $T(n)$ of the recurrence relation $$ p\ T(n-1) - \left[p+q+\overline{S} + \varepsilon \tilde{S}(n)\right]T(n) + q\ T(n+1) = 0,\\ ...
0
votes
0answers
28 views

Graphical intuition of function multiplication?

Reading up on convolution and though something that I haven't properly thought this far, What is the general heuristic for visualizing what the product of two or more function looks like? E.g. ...
3
votes
3answers
82 views

how to estimate $\prod_{k=2}^n \log(k)$?

I wonder if I can estimate $\prod_{k=2}^n \log(k)$ as $a^l$ for some a. I know that it is bounded by $e^{n^2}$, but I would like to get something finer.
0
votes
2answers
33 views

Proving $\prod_{i=2}^{i=n} \left(1-\frac{1}{i^2}\right) = \frac{n+1}{2n}$ by induction

So I have to prove the following using induction. ${\displaystyle \prod_{i=2}^{i=n} \left(1-\frac{1}{i^2}\right)} = \frac{n+1}{2n}$ I showed the basis step that if $n=i=2$, then the two functions ...
0
votes
2answers
59 views

Find the product of $(1+\frac{1}{5^{2^n}})$ for $n=0$ to $6$

i.e. $(1+\frac{1}{5})(1+\frac{1}{25})(1+\frac{1}{625})...(1+\frac{1}{5^{64}})$ I have no idea or clue to this problem. Is there a general trick/procedure to this kinds of problem?
0
votes
0answers
53 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 ...
4
votes
2answers
138 views

Good example showing why limits must exist in limit product rule

I'm looking for a way to show my calc 1 students not to use the limit laws without knowing that the individual limits exists. I could use $$\lim_{x\to 0} x^{2} \sin(1/x),$$ but by doing it wrong, one ...
0
votes
1answer
38 views

For $n \in \mathbb{N}$, find and prove a formula for $\sum_{i=1}^n \frac{1}{i(i+1)}$, plus related question.

I was fairly easily able to obtain and prove this formula for the sum: $$S(n)=\frac{n}{n+1}$$ by typical means of computing the partial sums, observing the pattern, and proving by induction. My ...
1
vote
1answer
27 views

$(C \times D) \cap (A \times B) \neq \emptyset \implies C \cap A \neq \emptyset$ and $D \cap B \neq \emptyset$

If $C$ and $D$ are open sets, and $(C \times D) \cap (A \times B) \neq \emptyset$, then why is it necessarily true that $C \cap A \neq \emptyset$ and $D \cap B \neq \emptyset$?
1
vote
1answer
92 views

Why $(\mathbb Q\times\mathbb Q)/(\mathbb Z\times{=})$ is not homeomorphic to $(\mathbb Q/\mathbb Z)\times(\mathbb Q/{=})$?

Let $\mathbb Q$ be the set of rationals with induced euclidian topology, let $\sim_1$ be the relation on $\mathbb Q$ which identifies all the integers, and let $\sim_2$ be the identity relation on ...
7
votes
1answer
402 views

Can we express sum of products as product of sums?

I've got an expression which is sum of products like: $$a_1 a_2 + b_1 b_2 + c_1 c_2 + \cdots,$$ but the real problem I'm solving could be easily solved if I could convert this expression into ...
4
votes
4answers
116 views

How is $0\cdot\infty= -1$?

It is known that the product of slopes of two perpendicular lines is equal to $-1$ ($m_1*m_2=-1$ for $m_1$ and $m_2$ being the slopes of the perpendicular lines $l_1$ and $l_2$). The slope of $x$-axis ...
2
votes
1answer
42 views

Average weighted by inverse distance to median equal to median?

Problem Statement I have a set of $N$ ordered elements such that $x = \{x_1, x_2, ..., x_q, x_p, ..., x_N\}$ where $x_q \le x_m \le x_p$ and $x_m$ is the median of the set $x$. I define a particular ...
0
votes
1answer
25 views

Equation with cos and product

How is it possible to prove that the multiplication from 0 to n of $$\cos{(x/2^k)}$$ with $x=\pi / 4$ is $$\dfrac{\sqrt{2}}{2} \times \dfrac{\sqrt{2+\sqrt{2}}}{2} \times \dfrac{\sqrt{2+\sqrt{2 + ...
0
votes
1answer
47 views

Is this infinite product for zeta(2) trivial?

I have crafted an infinite product for zeta(2) shown here. Euler's prime product is the only one I'm aware of. In checking Math World, I don't see any products. Is that because they are trivial?
0
votes
1answer
46 views

Solving for $n$ in this equation

I have this next equation: $c=\left( p^n\cdot c_0\right)\cdot \prod_{j=0}^{n-1}f(j)$ where: $c, p, c_0$ are known constants. $n$ is to be determined from this equation. The function $f(j)$ is ...
9
votes
1answer
81 views

Prove that a given expression is always an integer [duplicate]

Given integers $x_1, x_2, \dotsc, x_n$, prove that the expression $$ \prod \limits_{1\leq i<j\leq n}\frac{x_i - x_j}{i-j} $$ is always an integer. I think induction should work, but I ...
1
vote
2answers
73 views

Conversion from sum of product to product of sum

I do not understand how did he convert from this to this. Source :http://cs229.stanford.edu/notes/cs229-notes1.pdf Page 18
2
votes
1answer
35 views

Is the value of $c$ in $\prod_{i=1}^{n}\frac{p_i}{p_i-1}<e^c \cdot (\log p_n) \cdot(1+\frac{1}{\log_2p_n})$ known?

I Recently read this paper by Rosser and Schoenfeld (http://projecteuclid.org/download/pdf_1/euclid.ijm/1255631807) In Theorem 8, corollary 1, they state: $$\prod_{i=1}^{n}\frac{p_i}{p_i-1}<e^c ...
2
votes
2answers
55 views

Mathematical formalism for the “dot product” of three vectors

I know that the dot product of two vectors is the sum of element-wise multiplication. Using pseudo-MATLAB notation: (x,y) = sum(x.*y). I'm interested in ...
0
votes
1answer
23 views

Calculating better value products.

The special promotion tins of 300g cost 0.80$. The soup can also be bought in larger tin of 500g that cost 1.12$. Is it better value to buy the 500g tin or the special promotion tin? Show your ...
5
votes
2answers
42 views

How to get to this equality $\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1$?

How to get to this equality $$\prod_{m=1}^{\infty} \frac{m+1}{m}\times\frac{m+x}{m+x+1}=x+1?$$ I was studying the Euler Gamma function as it gave at the beginning of its history, and need to ...
3
votes
1answer
89 views

Question about Meijer-G definition and identity

I'm trying to wrap my mind around computation involving the Meijer $G$ function, as defined here. (Edit: I'm actually using a somewhat mixed notation using the definitions from MathWorld and the ...
3
votes
2answers
79 views

Prove that $\frac{1}{1999} < \prod_{i=1}^{999}{\frac{2i−1}{2i}} < \frac{1}{44}$

Prove that $$\dfrac{1}{1999} < \prod_{i=1}^{999}{\dfrac{2i−1}{2i}} < \dfrac{1}{44}$$ from the 1997 Canada National Olympiad. I have been able to prove the left half of the inequality using ...
3
votes
2answers
221 views

Matrix product notation

My lecturer has used some notation that I've never seen before: it is a (matrix) product symbol with a left-to-right arrow over the top. Does anybody know what this means? Thanks in advance. Edit: ...
5
votes
2answers
164 views

Closed form for $(2^1-1)(2^2-1)…(2^k-1)$?

Is there closed form for $\prod_1^{i=k}(2^i-1)$ ? I found that it is the product of the terms of the following arithmetico-geometric sequence : $$\{u_1=1,u_{n+1}=2u_n+1\}$$ I found nothing with ...
2
votes
1answer
36 views

If $\ne: X \times X \to S$ is continuous, is X hausdorff?

The Sierpiński space is defined like so: $$S = (\{\top, \bot\}, \{\emptyset, \{\top\}, \{\top, \bot\}\})$$ (A nice way to visualize is to take [0, 1], and glue 0 on $\bot$ and (0,1] onto $\top$.) ...
3
votes
0answers
44 views

Polynomial products [duplicate]

This problem $$ \large \displaystyle\prod \limits^{14}_{k=1}\cos \left( \frac{k \pi }{15} \right) =\ ? $$ I solved it in this way $$ x = \displaystyle \prod \limits^{14}_{k=1}\cos \left( \frac{k\pi ...
-1
votes
1answer
65 views

Difficult product problem $\prod \limits^{2014}_{k=1}\left( 1-\frac{1}{k^{2}} \right)$ [duplicate]

Evaluate the product $$\prod \limits^{2014}_{k=1}\left( 1-\frac{1}{k^{2}} \right)$$ Any help will appreciated!
6
votes
2answers
121 views

Calculate $\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$

Calculate $$\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$$ I can only bound it as follows: $$\binom{n}{i}<\left(\dfrac{n\cdot e}{k}\right)^k$$ $$\sum_{i = ...
5
votes
2answers
279 views

Inequality involving a product over the primes

Is there someone who is able to prove the following statement? $$\prod_{m=1}^n \dfrac{p_m-1}{p_m} \leq \dfrac{1}{\ln(n)}$$ for all integers $n >1$ where $p_m$ is the $m$-th prime number.
2
votes
0answers
28 views

Existence of formulae for sines/cosines of products of angles in terms of sines/cosines of original angles? [duplicate]

There was something that I was getting a little curious about. We know that there are the so-called compound-angle formulae for calculating sines and cosines of sums of angles in terms of those of the ...
1
vote
1answer
36 views

Evaluate products and sums

Did I evaluate the following terms correctly? Does the set notation in example b) allow me to chose the order of the terms? $$ a) \sum_{i=1}^6 ix^{i+1} = x^2+2x^3+3x^4+4x^5+5x^6+6x^7 \\ b) \prod_{i ...