# product

1. ### Compute infinite product

Compute \prod_{n=1}^{\infty} (1-\frac{2}{n^2 }).
2. ### tensor: outer product, representation, decomposition

It is given a tensor: $T=\begin{pmatrix} 1\\ 1 \end{pmatrix}\circ \begin{pmatrix} 1\\ 1 \end{pmatrix}\circ\begin{pmatrix} 1\\ 1 \end{pmatrix}+\begin{pmatrix} -1\\ 1 \end{pmatrix}\circ\begin{pmatrix} 1\\ -1 \end{pmatrix}\circ\begin{pmatrix} -1\\ 1 \end{pmatrix}$ 1) Why is it possible to write...
3. ### Book about Summation and Product of Sequences

Hello, I would like to know if there is any book devoted to summations and product of sequences (pi notation). With theory and good exercises.
4. ### Infinite product

How to show whether the infinite product converges ? p=\prod_{n=1}^{\infty } \prod_{m=1}^{\infty } \frac{n}{10^{m}} \; \; , n,m \in \mathbb{N} . Is p the product of all numbers in interval (0,1) ?
5. ### Show this example is an inner product

$B(x,y)=2\left( \sum_{i=1}^{n}x_{i}y_{i}\right)-\sum_{i=1}^{n-1}(x_{i}y_{i+1}+x_{i+1}y_{i})$ I need help showing that this is positive definite as i think I have already shown it is symmetric and linear in the first variable. Thanks guys!
6. ### Orthonormal basis B:{e1,e2,e3} with respect to an inner product space

We have the inner product <(x_1,x_2,x_3),(y_1,y_2,y_3)>=3x_1y_1+x_1y_3+y_1x_3+x_2y_2+2x_3y_3 I'm asked to find the orthonormal basis of R^3 that is given from the normal basis B=(e_1,e_2,e_3), e_1=(1,0,0), e_2=(0,1,0), e_3=(0,0,1) with respect to the above inner product I guess I should...

11. ### Product topology

why is (-1/n ,1/n) not an open in the product topology but open in the box topology?
12. ### simple true/false regarding inner product spaces

hello, could you check if i marked the correct answers? a)let v=M_{nxn}^R and let A, B \in V, (A,B) = tr(BA) defines an inner product on V. (incorrect, it does not and when i tried to use various inputs got also wrong answer. just not true) b)let V=R^2 and let u=(x_1,x_2) and...
13. ### Dot Product

Hey guys, if you would like to take a look here is a lesson I made on the Dot Product between vectors, I will explain what the dot product is, its properties and show you a step-by step numerical example. iRWTtPx-iyg On the channel you can find lessons on several other topics as well.
14. ### properties of inner product space

hello everyone! i am having a question and a problem regarding the properties of an inner product space. the question is: if a= \begin{pmatrix}4&1\\ 1&5\end{pmatrix}. then if we say v=(y1,y2),u=(x1,x2), the inner product in the standart base based on A would be (u,v)=4x1y1+x1y2+x2y1+5x2y2...
15. ### Derivatives: Chain Rule - Marginal Revenue Product

I am having trouble with part c of this question. I have already solved A and B but if someone could help me with part c I would greatly appreciate it. A factory owner who employs m workers finds they produce q=1.2m(1.2m+46)^3/2 units of product per day The total revenue R in dollars is R=...
16. ### Average product of labor.

APL = Q/L When I started to think that I have finally gotten the hang of this, I lose everything.... Help needed.
17. ### Opposite results of a cross product

In the usual right-hand rectangular coordinate \mathbf{i}_1\mathbf{i}_2\mathbf{i}_3 in 3D space, I tried to calculate \mathbf{i}_1\times\mathbf{a} where \mathbf{a}=a_1\mathbf{i}_1+a_2\mathbf{i}_2+a_3\mathbf{i}_3. If I use determinant method, I got \mathbf{i}_1\times\mathbf{a}= \left|...
18. ### Trigonometric Product Challenge

Prove for $m=2,3,...$ $$\sin\frac{\pi}{m}\sin\frac{2\pi}{m}\sin\frac{3 \pi}{m}\cdots\,\sin\frac{(m-1)\pi}{m}=\frac{m}{2^{m-1}}$$
19. ### split multiplication in integral

Hi, I've got an equation that looks like this: G = \int_0^X 4\pi (g-1) r^2 \left( 1-\frac{3r}{4R} + \frac{3r^3}{16R^3} \right) dr Now I would like to separate this integral into G = \int_0^X 4\pi (g-1) r^2 dr + Y or G = \int_0^X 4\pi (g-1) r^2 dr * Y where Y does not contain g. Is...
20. ### Trigonometric Product

If $\displaystyle \theta = \frac{2\pi}{2009},$ Then $\displaystyle \cos \theta \cdot \cos 2\theta\cdot \cos 3 \theta \cdot\cdots \cos 1004 \theta$ is