1. I

    Compute infinite product

    Compute \prod_{n=1}^{\infty} (1-\frac{2}{n^2 }).
  2. N

    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. J

    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. I

    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. J

    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. V

    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...
  7. M

    Element in a tensor product

    Let $p$ be a prime number and $M_p$ be the $Z$-module defined by $M_p=A_p/\mathbb Z$ where $A_p=\left\{a\in \mathbb Q\;|\;p^na\in \mathbb Z\;\text{for some}\; n\ge 0\right\}$. Let $A$ be a $\mathbb Z$-module. Every element of $A\otimes_{\mathbb Z}M_p$ may be expressed in the form $a \otimes(...
  8. J

    Interpreting Scalar Product (formula in equations), Please help!

    Hello all, I have recently been advancing my knowledge of mathematics by working through worksheets online. However, I am stumped at these particular questions, and have no clue where to begin and answer! Any chance of any answers? Answers would be appreciated as I can work through the steps...
  9. V

    Number of customers who have activated a product within 3 months of their purchase

    0 down vote favorite 1) I have a list of customers who have bought a product and consumed it on a certain date. I receive every month an excel sheet with the following columns: Customer_Account / Activation_Date / Country / 2) I have another excel sheet with the volume of sales...
  10. J

    Limit of Product of iid Random Variables

    Let $X_1, X_2, ... , X_n$ be i.i.d random variables with common density function: $f(x)=\frac{\alpha}{x^{\alpha+1}}I(x\geq 1)$ where $I(x\geq 1)$ is the indicator function and $\alpha >0$. Now define: $$S_n=\left[ \prod_{i=1}^{n}X_i \right ]^{n^{-1}}$$ Prove that $\lim_{n\rightarrow...
  11. A

    Product topology

    why is (-1/n ,1/n) not an open in the product topology but open in the box topology?
  12. M

    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. R

    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. M

    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. C

    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. V

    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. Z

    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. greg1313

    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. F

    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. P

    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