query: Elaborate definition of $\prod_{i} X_i$ where $X_i \in \left \{X_i \right \}_{i \in I}$
positive_passages.docid: 56440
positive_passages.text: Cartesian products and families
negative_passages.docid: 147111
negative_passages.text: Where is the Proof for the Sum-to-Product rule derrived from?
negative_passages.docid: 13864
negative_passages.text: $\prod_{k=0}^{\infty} \cos(x \cdot 2^{-k}).$
negative_passages.docid: 48544
negative_passages.text: Why is cross product defined in the way that it is?
negative_passages.docid: 81973
negative_passages.text: Definition of $d (P (x ,y )dx)$
negative_passages.docid: 35855
negative_passages.text: Closed form for an infinite product
negative_passages.docid: 132532
negative_passages.text: Things around $(\forall x)(\Psi \Rightarrow \Phi) \Leftrightarrow ((\exists z)\Psi) \Rightarrow \Phi)$
negative_passages.docid: 96366
negative_passages.text: show that for any n $\in$ $\Bbb N$ , if n $\ge$ 2, then $\prod_\limits{i=2}^n $ $\left( 1-\frac{1}{i^2}\right)$ = $\frac{n+1}{2n}$
negative_passages.docid: 83280
negative_passages.text: Infinite product $ \prod^{\infty}_{n=0}\left(1-\frac{1}{2^{3n}}-\frac{1}{2^{6n}}\right)$
negative_passages.docid: 14713
negative_passages.text: $X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology Then
negative_passages.docid: 97119
negative_passages.text: Prove that $\prod_{i=1}^n(1+x_i)\leq \sum_{i=0}^n\frac{S^i}{i!}$, where $x_i\in\mathbb{R^+}$.
negative_passages.docid: 101320
negative_passages.text: How to tell $i$ from $-i$?
negative_passages.docid: 109423
negative_passages.text: Purpose of universal property of a product.
negative_passages.docid: 158357
negative_passages.text: Infinite Product $\prod_{n=1}^\infty\left(1+\frac1{\pi^2n^2}\right)$
negative_passages.docid: 87636
negative_passages.text: A Product function for matrix products
negative_passages.docid: 118471
negative_passages.text: Prove that $1-\sum_{i=1}^n a_i < \prod_{i=1}^n(1-a_i) $
negative_passages.docid: 154837
negative_passages.text: mapping properties of $(1−z)^i$