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

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21
votes
4answers
1k views

What is to geometric mean as integration is to arithmetic mean?

The arithmetic mean of $y_i ... y_n$ 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 ...
21
votes
2answers
569 views

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 ...
19
votes
3answers
394 views

Finding $ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$

I would appreciate if somebody could help me with the following problem. How can we find the product $$ \prod_{n=1}^{999}\sin\left(\frac{n \pi}{1999}\right)$$
18
votes
3answers
446 views

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: $$\prod_{k=1}^j \frac{2 k}{j+k-1} = \frac{2^j j!}{(j)_j} \implies$$ ...
17
votes
4answers
2k views

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 $$\prod\limits_{n=2}^{\infty} \biggl(1-\frac{1}{n^{2}}\biggr)$$ ...
16
votes
3answers
821 views

A “fast” way for computing $ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $?

Which is the fastest paper-pencil approach to compute the product $$ \prod \limits_{i=1}^{45}(1+\tan i^\circ) $$
16
votes
0answers
418 views

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 $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty ...
15
votes
2answers
2k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
13
votes
5answers
604 views

Evaluating the infinite product $\prod_{k=2}^{\infty }\left ( 1-\frac{1}{k^2} \right ).$

Evaluate $$\lim_{ n\rightarrow\infty }\prod_{k=2}^{n}\left ( 1-\frac{1}{k^2} \right ).$$ I can't see anything in this limit , so help me please.
13
votes
1answer
298 views

A question about $\prod_{x\in \mathbb{R}^{*}}{x}$

When I was an elementary school student, I encountered some puzzles like "What's product of numbers of hair of each people in the world?", which answer was zero, because there's people with no ...
13
votes
2answers
920 views

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 ...
13
votes
0answers
410 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 ...
12
votes
2answers
389 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
12
votes
1answer
494 views

How does one calculate the product of $\tan 1^{\circ} … \tan 45^{\circ}?$

I have seen a question asked on yahoo asking to find the value of $\tan 1^{\circ} \cdot \tan 2^{\circ} \cdot \dots \cdot \tan 45^{\circ}$ (in degrees) I have seen various results concerning ...
11
votes
2answers
672 views

Infinite products - reference needed!

I am looking for a small treatment of basic theorems about infinite products ; surprisingly enough they are nowhere to be found after googling a little. The reason for this is that I am beginning to ...
10
votes
3answers
264 views

Product of two algebraic varieties is affine… are the two varieties affine?

Let $X_1$ and $X_2$ two algebraic varieties such that their product $X_1\times X_2$ is affine. Are $X_1$ and $X_2$ affine then? If this is not true, could you give a counterexample?
10
votes
2answers
235 views

Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

The current issue (vol. 120, no. 6) of the American Mathematical Monthly has a proof by probabilistic means that $$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k} $$ for ...
10
votes
1answer
247 views

How to prove that $\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}(1-\frac{x^2}{n^2})$ [duplicate]

How to prove that $$\frac{\sin \pi x}{\pi x}=\prod_{n=1}^{\infty}\left(1-\frac{x^2}{n^2}\right)$$ I tried it with the Taylor series of $\sin(x)$ but I failed. Is there any help?
9
votes
4answers
1k views

Evaluating the product $\prod\limits_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)$

Recently, I ran across a product that seems interesting. Does anyone know how to get to the closed form: $$\prod_{k=1}^{n}\cos\left(\frac{k\pi}{n}\right)=-\frac{\sin(\frac{n\pi}{2})}{2^{n-1}}$$ I ...
9
votes
3answers
376 views

Prove this product

How to prove this product? $$\prod\limits_{k=2}^ n {\frac{k^2+k+1}{k^2-k+1}}=\frac{n^2+n+1}{3}$$
8
votes
7answers
569 views

Is there a sequence in $(0,1)$ such that the product of all its terms is $\frac{1}{2}$?

I am trying to construct a sequence $\{x_{n}\} \in (0,1)$ such that such that the product of all its terms is $\frac{1}{2}$. Please can I have any clue to solve my problem? Thanks.
8
votes
2answers
217 views

How to find finite trigonometric products

I wonder how to prove ? $$\prod_{k=1}^{n}\left(1+2\cos\frac{2\pi 3^k}{3^n+1} \right)=1$$ give me a tip
8
votes
2answers
183 views

Is $ \prod\limits_{k=0}^\infty \left(1 + \frac{1}{k!}\right) = \mathrm e^2 $?

I was playing around and I came up with this product, which I believe to be equal to $\mathrm e^2$. $$ \prod_{k=0}^\infty \left(1 + \frac{1}{k!}\right) \stackrel{?}{=} \mathrm e^2 $$ After ...
8
votes
0answers
177 views

How to find the the product $\left(1 - \frac{1}{a}\right)\left(1-\frac{1}{a^{2}}\right)\left(1-\frac{1}{a^{3}}\right)\ldots$ [duplicate]

Possible Duplicate: Result of the product $0.9 \times 0.99 \times 0.999 \times \dots$ How to find the product $$\left(1 - ...
7
votes
5answers
638 views

Definition of the Infinite Cartesian Product

(1) If $X$ and $Y$ are two sets, we define the Cartesian product $X \times Y$ as the set of ordered pairs $(x,y)$, such that $x \in X$ and $y \in Y$. (2) On the other hand [Folland, Real Analysis, ...
7
votes
1answer
395 views

Evaluation of a product of sines [duplicate]

Possible Duplicate: Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$ I am looking for a closed form for this product of sines: \begin{equation} \sin ...
7
votes
2answers
400 views

Proving an infinite product formula

I have found this formula and I am trying to prove it , but I have not any idea how to deal with it: $$e^{ax}-e^{bx} = ...
7
votes
1answer
134 views

Zeros in the complex plane and convergence

I'm doing some number theory which requires some work in $\mathbb{C}$, but unfortunately my complex analysis is a little rusty. A text I am reading states the following: ...and given that ...
6
votes
4answers
114 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 ...
6
votes
3answers
205 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 ...
6
votes
3answers
180 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: $$ S_n := (n\prod_{k=2}^{n-1} k^{2^{n-k}})^{2^{-k}} $$ But I don't even quite know if I'm on the right track. If I ...
6
votes
1answer
275 views

Generalization of the series for $\frac{\pi^2}{6}$? Is there a more elementary proof?

In the same vein as: $ \frac{\pi ^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{25} \cdots $ Starting with: $ \displaystyle \prod_{n=1}^{\infty} \left( 1 -\frac{q^2}{n^2} \right) = ...
6
votes
2answers
666 views

How was Euler able to create an infinite product for sinc by using its roots?

In the Wikipedia page for the Basel problem, it says that Euler, in his proof, found that $$\begin{align*} \frac{\sin(x)}{x} &= \left(1 - \frac{x}{\pi}\right)\left(1 + \frac{x}{\pi}\right)\left(1 ...
6
votes
3answers
123 views

closed-form expressions for product of 3n+k where k = 1 or 2

There are some easy products that can be written in closed form in terms of factorials: $ 2 \times 4 \times 6 \times ... 2n = n! \times 2^n$ $ 1 \times 3 \times 5 \times ... (2n-1) = {{(2n)!} ...
6
votes
1answer
192 views

Which is the Abel's theorem invoked in the context of convergence of this infinite product?

Motivation: As I wrote in this answer the following product is evaluated in Proposition 5 of Jean-Paul Allouche and Jeffrey Shallit's paper The ubiquitous Prouhet-Thue-Morse sequence ...
6
votes
1answer
127 views

Is there another way to write the product $\prod_{k=0}^n\left(k+\alpha\left(-1\right)^{k+1}\right)$?

I have the following expression $$ \prod_{k=0}^n \left(k + \alpha(-1)^{k+1}\right), $$ which is, for example, $(0-\alpha)(1+\alpha)(2-\alpha)$ for $n = 2$. Is there a way to write this using ...
6
votes
1answer
218 views

Uncountable product in the category of metric spaces.

I need to prove that category $\mathrm{Met}$ of metric spaces and continuous maps doesnt possess uncountable product of non-one point spaces. Definition. A pair $(X,\{\pi_\nu:\nu\in\Lambda\})$ where ...
5
votes
5answers
1k views

Can the limit of a product exist if neither of its factors exist?

Show an example where neither $\lim\limits_{x\to c} f(x)$ or $\lim\limits_{x\to c} g(x)$ exists but $\lim\limits_{x\to c} f(x)g(x)$ exists. Sorry if this seems elementary, I have just started my ...
5
votes
5answers
88 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 ...
5
votes
1answer
302 views

When is $\displaystyle \prod_i \prod_j a_{i} a_{j} = \Bigl(\prod_i a_i\Bigr)^2$

In statistical mechanics, I used to use the procedure that if $a_{ij}=a_i a_j$ $$\prod_i\; \prod_j a_{i}a_{j} = \biggl(\prod_i a_i\biggr)\vphantom{\Bigr)}^2$$ However, today I noticed, $$\prod_i\; ...
5
votes
3answers
83 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 ...
5
votes
2answers
54 views

Evaluating $\prod_{r=1}^{n} (2r+1)$

Could someone please help me as to how I'd go about evaluating: $$\prod_{r=1}^{n} (2r+1)$$ I have that written out, it is: $$1 \cdot 3 \cdot 5 \cdots (2n-1) \cdot (2n+1)$$ furthermore: ...
5
votes
2answers
213 views

Is there a known closed form number for $\prod\limits_{k=2}^{ \infty } \sqrt[k^2]{k}$

$f(x)=\sum\limits_{k = 2 }^ \infty e^{-kx} \ln(k) $ $\int\limits_0^{\infty}\int\limits_x^{\infty}\, f(\gamma)\, d\gamma dx=\sum\limits_{k = 2 }^ \infty \frac{1}{k^2} \ln(k) $ ...
5
votes
1answer
269 views

Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?

If $a_1\ge a_2 \ge a_3 \ldots $ and if $b_1,b_2,b_3\ldots$ is any rearrangement of the sequence $a_1,a_2,a_3\ldots$ then for each $N=1,2,3\ldots$ one has $$\sum^N_{n=1}\left(\prod_{i=1}^n b_i ...
5
votes
1answer
123 views

Evaluate $\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$

Difficult question from some test somewhere (I forget). $$\prod_{x=2}^\infty\frac{x^4-1}{x^4+1}$$ $x$ is, of course, an integer.
5
votes
2answers
619 views

Proving the AM:GM inequality

I am doing past exam papers preparing for the finals and I came across this questions about three times: Prove that: $$\frac{a_{1}+a_{2}+\cdots+a_{n}}{n}\geq \sqrt[n]{a_{1}.a_{2}...a_{n}}$$ ...
5
votes
1answer
167 views

Continuous maps from products of topological spaces

Let $X,Y,Z$ be topological spaces. It is well-known that if $F:X\times Y\to Z$ is a continuous map, we can define a map $$\overline{F}:X\to C(Y,Z) \\\overline{F}(x)(y)=F(x,y)$$ where $C(Y,Z)$ is the ...
5
votes
1answer
110 views

Modulus of infinite product of complex functions

We know that for complex (entire) functions $f,g$ we have $|f(z)g(z)|=|f(z)||g(z)|$, where $|.|$ means complex modulus. What about if we have an infinite product? Is it true that $$\bigg| ...
4
votes
5answers
795 views

The limit of infinite product

Is there any elementary proof that the limit of infinite product $\prod_{k=1}^{\infty}(1-\frac{1}{2^k})$ is not 0?
4
votes
2answers
566 views

How to evaluate $\lim\limits_{n\to+\infty} \prod\limits_{k=1}^n (1+k/n^2)$?

I've got a limit which puzzle me several days. The question is $$ \lim_{n\to+\infty} \prod_{k=1}^n\left(1+\frac{k}{n^2}\right).$$ Can you help me? Thank you in advance