# Tagged Questions

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

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### How I can calculate this product

How I can calculate this product: $$s=∏_{j=1}^{p-3}\sinh^2[2^{j-1}\cosh^{-1}( 2)]$$ for a natural number $p>3$.
57 views

### $S$ nonempty finite subset of group $G$ for which $SS=S$. $S$ is subgroup.

Let $S\subset G$, $S$ finite and nonempty, $G$ group. Suppose additionally that $$SS=\{s_1 s_2: s_1\in S, s_2 \in S\}=S.$$ How can I prove that $S$ is a subgroup of $G$? Does this hold for $S$ ...
302 views

### Other proof for tensor product equal to zero (without use Nakayama lemma and Jacobson radical)

Let $R$ is commutative local ring, $M$ is $R$-module, $N$ is $R$-module. If $M,N$ are finitely generated, how to prove: $M \otimes_R N=0$ if and only if $M=0$ or $N=0$. If we delete the ...
264 views

### Why is empty product defined to be $1$? [duplicate]

For example $\prod_{2 \le j < 1} 2^j= 1.$ How does that happen?
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### Counterexamples in $R$-modules products and $R$-modules direct sums and $R$-homomorphisms (Exemplification)

Q: (Exemplification) Do examples family of $R$-modules like $\{M_i\} , \{N_i\}$ and $R$-modules like $M,N$ such that every of four question in underline is true. (in other word for every question ...
106 views

### Idea for primality testing based on a trigonometric product

This is an idea that I had about 3 months ago. I tried some college professors, they didn't care. I tried to solve, but with no luck. I ask for help to find the closed form of the following product ...
123 views

1k views

### Theorem? For any sets A, B, C, and D, if A x B is a subset of C x D then A is a subset of C and B is a subset D.

Is the following proof correct? If so, what proof strategies does it use? If not, can it be fixed? Is the theorem correct?   Proof. Suppose A x B is a subset of C x D. Let a be an arbitrary element ...
165 views

### The limit of products of the form $(n^3-1)/(n^3+1)$

Calculate $$\lim_{n \to \infty} \frac{2^3-1}{2^3+1}\times \frac{3^3-1}{3^3+1}\times \cdots \times\frac{n^3-1}{n^3+1}$$ No idea how to even start.
324 views

### How to compute the following integral in $n$ variables?

How can the following integral be calculated: $$I_n=\int_0^1\int_0^1\cdots\int_0^1\frac{\prod_{k=1}^{n}\left(\frac{1-x_k}{1+x_k}\right)}{1-\prod_{k=1}^{n}x_k}dx_1\cdots dx_{n-1}dx_n$$ There should ...
53 views

### How to calculate the product of a set

How can you calculate the product of a set $A$, denoted by $\Pi A$ and defined by $\forall z \in \Pi A(z \subseteq \bigcup A \wedge \forall y \in A (\exists x (z \cap y = \lbrace x \rbrace)))$ ...
209 views

### Product of ergodic transformations

I'm asked to give an example, that the product of two ergodic systems is not ergodic in general. I know that for $X_1=X_2=(S^1,B,m,R_a)$ (the irrational rotation on the unit circle with Lebesgue ...
187 views

### relationship between multiplication and correlation

is there a deep interpretation of multiplication as correlation? is this in some sense the most fundamental way to "combine" objects (eg numbers) into products? my reasons for asking are that the ...
How can I simplify the following product of two sums: $$\biggl(\, \sum ^{n}_{k=0}a_{k}\biggr) \biggl(\, \sum ^{n}_{k=0}\dfrac {1}{a_{k}}\biggr)$$