Questions tagged [products]

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

A finite product is an object of the form $$ \prod _{k=1}^{n} a_k = a_1 \times a_2 \times \ldots \times a_n $$The $a_k$ are often complex numbers or functions. For finite products the only issue of convergence is that none of the $a_k$ are identically zero.

Examples of finite products include:

  • The (perhaps misnamed) Fundamental Theorem of Algebra asserts that every polynomial in $\mathbb{C}[x]$ can be written as $$ p(x) = c \prod _{k=1}^n (x-\alpha_k), $$where $c$ is a constant and $\alpha_k$ are the roots of $p$ listed with multiplicities.
  • The falling factorial $(x)_n$ is $$ (x)_n = \prod_{k=0}^{n-1}(x-k) $$
  • The Lagrange interpolation formula: let $\{(x_i,y_i)\}_{i=0}^{n}$ be a set of $n+1$ points with the $x_i$ distinct. The unique polynomial of degree at most $n$ $P_n(x)$ passing through them is given by $$ L_i(x)=\prod\limits_{\substack{k=0 \\ k\neq i}}^n \frac{(x-x_k)}{(x_i-x_k)} $$ $$ P_n(x) = \sum_{k=0}^{n} y_i L_i(x) $$
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Is there an "inverted" dot product?

The dot product of vectors $\mathbf{a}$ and $\mathbf{b}$ is defined as: $$\mathbf{a} \cdot \mathbf{b} =\sum_{i=1}^{n}a_{i}b_{i}=a_{1}b_{1}+a_{2}b_{2}+\cdots +a_{n}b_{n}$$ What about the quantity? $$\mathbf{a} \star \mathbf{b} = \prod_{i=1}^{n}…
doc
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Is $0! = 1$ because there is only one way to do nothing?

The proof for $0!=1$ was already asked at here. My question, yet, is a bit apart from the original question. I'm asking whether actually $0!=1$ is true because there is only one way to do nothing or just because of the way it's defined.
Ali Abbasinasab
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Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
Ali_ilA
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Does multiplying all a number's roots together give a product of infinity?

This is a recreational mathematics question that I thought up, and I can't see if the answer has been addressed either. Take a positive, real number greater than 1, and multiply all its roots together. The square root, multiplied by the cube root,…
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When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty \# = \prod_{k=1}^\infty p_k = 4\pi^2$$ where $n\#$ is a primorial, and $p_k$…
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Finding Value of the Infinite Product $\prod \Bigl(1-\frac{1}{n^{2}}\Bigr)$

While trying some problems along with my friends we had difficulty in this question. True or False: The value of the infinite product $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ is $1$. I couldn't do it and my friend justified…
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What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i \ldots y_n$ is: $$\frac{1}{n}\sum_{i=1}^n~y_i $$ For a smooth function $f(x)$, we can find the arithmetic mean of $f(x)$ from $x_0$ to $x_1$ by taking $n$ samples and using the above formula. As $n$ tends to infinity, it…
netvope
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Why is the cartesian product so categorically robust?

The major "broad/natural" categories I encounter in daily life are: sets, groups, topological spaces, smooth manifolds, vector spaces over a fixed field $k$, $k$-schemes, rings, $A$-algebras for a fixed commutative ring $A$, and $A$-modules. I think…
Ben Blum-Smith
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Product of cosines: $ \prod_{r=1}^{7} \cos \left(\frac{r\pi}{15}\right) $

Evaluate $$ \prod_{r=1}^{7} \cos \left({\dfrac{r\pi}{15}}\right) $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] $$ but I couldn't find the product. Any help will…
Henry
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How to prove those "curious identities"?

How to prove $$ \prod_{k=1}^{n-1} \sin\left(\frac{k\pi}{n}\right) = \frac{n}{2^{n-1}}$$ and $$ \prod_{k=1}^{n-1} \cos\left(\frac{k\pi}{n}\right) = \frac{\sin(\pi n/2)}{2^{n-1}}$$
Victor
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Why is there no explicit formula for the factorial?

I am somewhat new to summations and products, but I know that the sum of the first positive n integers is given by: $$\sum_{k=1}^n k = \frac{n(n+1)}{2} = \frac{n^2+n}{2}$$ However, I know that no such formula is known for the most simple product (in…
zerosofthezeta
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Trying to prove that $\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$

How could one prove that: $$\sum_{j=2}^\infty \prod_{k=1}^j \frac{2 k}{j+k-1} = \pi$$ This is about as far as I got: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ $$\sum_{j=2}^\infty\frac{2^j j!}{(j)_j} = \frac{2^2 2!}{(2)_2} +…
JohnWO
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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 something like : $$(a_1+b_1+c_1+\cdots) \cdot…
lavee_singh
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Compute $\lim\limits_{n\to\infty} \prod\limits_2^n \left(1-\frac1{k^3}\right)$

I've just worked out the limit $\lim\limits_{n\to\infty} \prod\limits_{2}^{n} \left(1-\frac{1}{k^2}\right)$ that is simply solved, and the result is $\frac{1}{2}$. After that, I thought of calculating $\lim\limits_{n\to\infty} \prod\limits_{2}^{n}…
user 1591719
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Is there any way to define arithmetical multiplication as other thing than repeated addition?

Is there any way to define arithmetical multiplication as other thing than repeated addition? For example, how could you define $a\cdot b$ as other thing than $\underbrace{a+a+\cdots+a}_{b \text{-times}}$ or $\underbrace{b+b+\cdots+b}_{a…
YoTengoUnLCD
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