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

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0answers
52 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
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
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1answer
95 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
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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 ...
0
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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.
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4answers
124 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
409 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
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0answers
116 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
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1answer
531 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
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.
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0answers
40 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 ...
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3answers
150 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
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 ...
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2answers
54 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
58 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
88 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 $X'$, ...
4
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2answers
79 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
73 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 + ...
0
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2answers
109 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 ...
0
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1answer
31 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
96 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 ...
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0answers
81 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). ...
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5answers
101 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 ...
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3answers
43 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
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1answer
46 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 ...
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4answers
100 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 ...
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2answers
35 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 ...
4
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2answers
98 views

A curious product formula

Fiddling with Mathematica seems to suggest the following: $$\frac{(2^2)(4^2)(6^2)\cdots(2N^2)}{(1^2)(3^2)(5^2)\cdots(2N-1)^2}=N\pi+\frac{\pi}{4}+\frac{\pi}{32N}-\frac{\pi}{128N^2}+o(1/N^2).$$ Does ...
0
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1answer
47 views

Proof of “factorization of polynomials” using only Complex Analysis

I ask for the proof of the following: If $p$ is a polynomial with degree $n\ge 1$ and zeros in $A\subseteq \mathbb C$ whose order (multiplicity) is given by $n:A\to \mathbb N^*$ then $A$ is finite ...
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0answers
96 views

Notation for Kronecker product of a matrix and itself?

What is the notation for the Kronecker product of a matrix and itself? In other words, is there a short-hand way I can express the following: $X⊗X$ $X⊗X⊗X$ $X⊗X⊗X⊗X$ Where $X$ is a matrix? What ...
2
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1answer
84 views

Is anyway to prove this: $\prod_{k=1}^{n}(a_{k})< (1/n^n)*(\sum_{k=1}^{n}(\sqrt{1+a_{k}*a_{k+1}}))^n$

$$ \prod\limits_{k=1}^{n}a_{k} < {1 \over n^{n}}\left(\,\sum_{k = 1}^{n}\,\sqrt{1+a_{k}\,a_{k+1}\,}\,\right)^n $$ ak and n are positive real number greater than 0. EDIT: a_{k+1} becomes a_{1} ...
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1answer
44 views

Is it generally preferred that empty products are gotten rid of where possible?

Is it generally preferred that empty products are gotten rid of where possible? For example: Stewart's structure theorem says that for a positive integer $n$, every positive integer $\leq n$ has a ...
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3answers
26 views

Products of Functions that Don't Depend on Index

Is there a general property of products that allows you to simplify $$ \prod_{i = 1}^n f(x) \, g(i) $$ where $f(x)$ does not depend on $i$? Would it just be $$ f^{n}(x) \prod_{i = 1}^n g(i) $$ ...
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0answers
32 views

Efficient Cartesian product which ignores classes of elements

Given $n$ sets $X_1,X_2,..,X_n$, and what I am calling an ignore set $I = \{I_1, I_2,..,I_m : \forall i \in I_i, i \in \bigcup X_i\}$. I would like to find the cartesian product $X_1 \times X_2 ...
0
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1answer
97 views

Proving the formula for the directional derivatives of the of the sum and dot product of two functions

Define the directional derivative of a function $\textbf{f}$ at $\textbf{c}$ in the direction $\textbf{u}$ by $$\textbf{f}\hspace{0.04in}'(\textbf{c};\textbf{u}) = \lim_{h \rightarrow 0} ...
2
votes
2answers
99 views

Finding $\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$

Recently got this on a test: $$\lim_{n\to\infty}\frac{\prod_{k=1}^n(2k-1)}{(2n)^n}$$ Because it's a freshman calculus course, I think we were expected to solve it like a physicist. Taking a look at ...
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1answer
42 views

Can we express these sets as Cartesian products of two subsets of $\mathbf{R}$?

Let sets $A$ and $B$ be given as follows: $$A := \{ (x,y) \in \mathbf{R}^2 | \ \ x < y \ \ \} $$ and $$B := \{ (x,y) \in \mathbf{R}^2 |\ \ x^2 + y^2 < 1 \ \ \}.$$ Can we express $A$ or $B$ as ...
2
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1answer
74 views

$\prod_{k=1}^{n-1}\cos\left(\theta+\frac{k\pi}{n}\right)$

I know that the first one of the following identities holds, yet I don't know the identity of the general case as shown in the title and the bottom of the page. Is there anyone who knows the closed ...
0
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1answer
38 views

Convergence of a product

I want to show the following product converges for $x<e$ and diverges for $x \ge e$: $$\lim_{n\to\infty}\prod_{i=0}^{n-1}\left(x-\frac{xi}{n}\right).$$ To do this, I would need to show that the ...
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0answers
34 views

Simplifying products

Sorry for the very general title, but I don't even know how to name my question. I got a formula which is: $f(n)=\prod_{i = 0}^{\infty} ((n \; \mathrm{rem} \; p^{i + 1}) \; \mathrm{div} \; p^i + 1) ...
1
vote
1answer
49 views

how is a factorial fraction equal to the product notation

how is the $\prod$(2k-3) from k=2 to n equal to : ${(2n-3)!\over 2^{n-2}(n-2)!}$ where n $>=$ 2 i know that the (2n-3)! is equal to the product of 2k-3 from k=2 to n but I can't figure out the ...
0
votes
1answer
40 views

Product of randomly drawn numbers

Here are two code line to run in R: prod(rnorm(100, mean=1, sd=0)) # (1) prod(rnorm(100, mean=1, sd=0.2)) # (2) $prod(..)$ returns the product of a sequence. The sequence it given by ...
6
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3answers
230 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
2
votes
2answers
52 views

Generalizing the Product Rule

How would I go about generalizing the product rule to the product of $n$ functions $\psi_1(x), \ \psi_2(x), ..., \ \psi_n(x)$? That is, I'm hoping to obtain an expression for $$ \frac{d}{dx} \prod_{j ...
2
votes
2answers
254 views

Product of two complementary error functions (erfc)

I believe that (i.e., it would be convenient if, and visually appears that) the product of the two complementary error functions: ...