# Tagged Questions

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

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### What is $\prod _{j=1}^n \left(\sqrt{j}+1\right)$?

By the Fundamental Theorem of Algebra, it is easily seen that for a monic polynomial $p(x) \in \mathbb{C}[x]$, $$\prod _{j=1}^n p(j) = \frac{\prod_{p(r)=0}\Gamma(1+n-r)}{\prod_{p(r)=0}\Gamma(1-r)},$$ ...
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### nth product of sequential matrices

$\forall n \in \mathbb{N}$, let: $$P_n = \left( \begin{matrix} a & 1-a \\ b_n & 1-b_n \end{matrix} \right).$$ Whereby $\{b_n\}_{n \in \mathbb{N}}$ is a monotonically increasing sequence of ...
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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 ...
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### what does $(A\cdot\nabla)B$ mean?

I was studying a physics book and I saw this expression $$(A\cdot\nabla)B$$ where $A$ and $B$ are vectors. What's the definition of this? I've also seen this in some identities
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### How to estimate the product of the $k$ largest eigenvalues of a matrix

Now I have a question which let me to prove that the product of the largest $k$ singular values of a real matrix is always larger than the one of $k$ largest eigenvalues. For $k=1$, I use the ...
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### Value of finite product based on empty set

How does one evaluate the following product if the set S happens to be empty? \begin{aligned} f(n)= n \prod_{x \in S} \left(1-\frac{1}{x}\right) \end{aligned} Is the value simply n or is it ...
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### Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$
I want to prove $$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$ if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$ I have no idea where to start, any advice would be greatly appreciated!