1
vote
1answer
79 views

If $a + b + c = 0$ prove that

If $a + b + c = 0$, prove that 1)$$ \sum_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 3 $$ 2)$$ \prod_{\text{cyc}}{\frac{4bc - a^2}{bc + 2a^2}} = 1 $$ There is a solution that uses two cubic ...
1
vote
0answers
107 views

Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$

For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate $n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$ So the product goes up to $k$ and I ...
1
vote
1answer
23 views

$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$

I have found this equality: $$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$$ Do you think is it true?
0
votes
1answer
85 views

A frightening sum [duplicate]

Let $x_1,\ldots,x_r,y_1,\ldots,y_p,z_0,\ldots,z_r,t_0,\ldots,t_p$ be complex numbers. Let $A$ be the ring generated by these numbers. Prove the following holds in $\mathbb C(A)$. ...
0
votes
0answers
32 views

How to derive finite formula for PRODUCT of terms from 1 to n?

should i use limits? I totally forgot how to work with them, i can imagine doing summation(sigma) but not product
0
votes
1answer
36 views

Summation in 104 Number Theory problems

There's a paragraph of 104 Number Theory problems, on page $9$ that says: From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$ using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
3
votes
3answers
134 views

How find this value $\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$

Find the value $$\prod_{k=1}^{\infty}\left(1+\dfrac{1}{k^5}\right)$$ I know this :How find this $\prod_{n=2}^{\infty}\left(1-\frac{1}{n^6}\right)$ and maybe can find the $2k+1$? can you someone konw ...
1
vote
1answer
51 views

Expanding a product formally.

Let $a_1,...,a_n$ be real numbers. I don't know how to formally expand the following product $$ \prod_{k=1}^n(1+a_k) $$ I'm guessing something like (edited) $$1+\huge\sum_{k=1}^n \; ...
0
votes
1answer
51 views

Multiplying Sigmas(sums)

I would be grateful if someone please rewrite or expand this please. I have problem multiplying two sigmas ($\sum $) $$ (d(n)-\sum_{k=-\infty}^{\infty} h_k x(n-k)) \times ...
0
votes
1answer
17 views

Probability and Production equation translation

I know that the pi is like a summation except multiplication instead of addition and that P(x) means the probability of, but I'm having trouble putting it all together, esp the $w_i$ such that $w_1, ...
0
votes
0answers
47 views

Nested sums and products

I am trying to determine how to express this series in a general form as a summation of products: \begin{equation} \begin{aligned} (p_{i1}p_{j1})(1-p_{i2}p_{j2})(1-p_{i3}p_{j3}) \mbox{ } &+ \\ ...
0
votes
2answers
44 views

Lagrange polynomials sum to one

I've been stuck on this problem for a few weeks now. Any help? Prove: $\sum_{i=1}^{n}\prod_{j=0,j\neq i}^{n}\frac{x-x_j}{x_i-x_j}=1$ The sum of lagrange polynomials should be one, otherwise affine ...
0
votes
1answer
34 views

Understanding relation between Product and Summation Notation

So I am given the following: $n = \sum_{i=1}^{k}m_{i}$ I am also given $x = \sum_{i=1}^{k}log(m_{i}) = log\prod_{i=1}^{k}m_{i}$ I was only given the first part, however I believe that is a ...
3
votes
1answer
86 views

Can Pi prod be expressed using Sigma Notation?

My question is simple (but difficult for me): $\prod(x)$ be expressed interms of $\sum (x)$
1
vote
2answers
122 views

How to interchange a sum and a product?

I have this expression: $$\sum_{\{\vec{S}\}}\prod_{i=1}^{N}e^{\beta HS_{i}}=\prod_{i=1}^{N}\sum_{S_{i}\in\{-1,1\}}e^{\beta HS_{i}} \qquad (1)$$ Where $\sum_{\{\vec{S}\}}$ means a sum over all possible ...
0
votes
2answers
48 views

swap summation and multiple

In which case can we swap summation and multiple? ie. $$\sum_{i=1}^n\prod_{j=1}^na_{ij}=\prod_{j=1}^n\sum_{i=1}^na_{ij}$$ if we can't swap like this, please tell me how can we swap them?
0
votes
0answers
44 views

Way to split up product of summation

If I have $\sum_{n=1}^{\infty}f(x)g(x)$, is there any way to split this up? Thanks.
0
votes
0answers
32 views

Can we convert a sum-of-products into strictly sums?

So I start with a sum-of-products: $$f = \sum_{k=a}^b{\prod_{j=c}^d{g(j,k)}}$$ I'm wondering if we can somehow convert this into a sum of sums, ie: $$f = \sum_{k=a}^b{\sum_{j=c}^d{h(j,k)}}$$ It's ...
0
votes
1answer
19 views

Exponential equivalent for geometric space

I'm just starting a foray into geometric algebra and calculus so that I can develop a geometric version of the standard arithmetic neural net. Specifically when calculating the error function for a ...
4
votes
2answers
91 views

Coefficients of $(x-1)(x-2)\cdots(x-k)$

I'm interested in the coefficients of $x$ in the expansion of, $$ (x-1)(x-2)\cdots(x-k) = x^k + P_1(k) x^{k-1} + P_2(k)x^{k-2} + \cdots + P_k(k),$$ Where $k$ is an integer. In particular I am ...
1
vote
2answers
51 views

What are the properties around Pi products

There are some well-known evaluations around summations like $\sum 1$ or $\sum i^2$ but what are these properties for products, specifically something like $\prod_{i=0}^{n-1} (n-i)$. I basically have ...
3
votes
0answers
74 views

Product of Summations for All Subsets

We have a set $X$ of $n$ integers $\{$$x_1$, $x_2$, .. , $x_n$$\}$, for which there are $2^n$ total subsets. The summation $s$ of a subset $X'$ is simply the sum of all integers present in $X'$, ...
1
vote
3answers
68 views

Finding limit of a product.

Prove:$$\lim_{n \to\infty }\frac{1}{n}\left[\prod_{i=1}^{n}(n+i) \right ]^{\frac{1}{n}}=\frac{4}{e}$$ I tried using Squeeze Theorem but can't go beyond $1<L<2$. $$\lim_{n\to\infty} \left( 1 + ...
2
votes
3answers
40 views

Sum and product of integers conserving order

I have the feeling this is true, but can't prove it: $$\sum_n^An\lt\sum_n^Bn\implies\prod_n^An\lt\prod_n^Bn$$ Where $A\subset\mathbb N-\{0, 1\},B\subset\mathbb N-\{0, 1\}$ Example: ...
0
votes
1answer
43 views

What's the maths symbol for alternation of product and sums?

Is there a mathematics symbol for referring to the equation below? (((((((((((100*y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x) *y-x = 0, x = 9.8 I've tried using capital ...
0
votes
2answers
33 views

How to repesent n x m multiplication into symbol notation?

I am not a mathematician and so I might not be using the right terms. I have a vector of n components and another vector of m components ...
0
votes
0answers
37 views

Product of consecutive integers as alternating sum

In the derivation of the Indian Buffet Process (Indian Buffet Process paper -- see page 1218), we have the following step: $\prod_{k=1}^{K_+}{(K-k+1)} = ...
6
votes
3answers
222 views

Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$

For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of ...
2
votes
0answers
56 views

Why does $\frac{d}{d\theta} \theta\prod_{i=1}^nx_i = \sum_{i=1}^nx_i$

Is this just the product rule? I have this in my notes but I didn't think anything of it and now I'm wondering how this happens? Edit: Im working with maximum likelihood estimation and in my notes I ...
1
vote
1answer
210 views

Summation of a product

I have basic doubt regarding summation over a product. The starting equation is $\sum_{z}\prod_{k=1}^{K}\pi_k^{z_k}f(x|\mu_k)^{z_k}$ and it is given that $\sum_{k}{z_k}=1$ How does it become ...
1
vote
1answer
64 views

Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
0
votes
2answers
105 views

How to go from a sum to a product and a product to a sum?

I have read here (third post down) that exponentials turn sums into products and logarithms turn products into sums. Can someone please further explain this?
10
votes
2answers
257 views

Prove $\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k}$ and more

The current issue (vol. 120, no. 6) of the American Mathematical Monthly has a proof by probabilistic means that $$\sum_{k=0}^n \binom{n}{k}(-1)^k \frac{x}{x+k} = \prod_{k=1}^n \frac{k}{x+k} $$ for ...
3
votes
1answer
164 views

Partition Proof

Let $\lambda$ be a partition of $N$ of rank $r$. How can I show that: $$\sum_wx^\lambda(w)=f^\lambda(-1)^{t(\lambda)}\prod^r_{i=1}(\lambda_i-1)!(\lambda'_i-1)!$$ where $w$ ranges over all ...
0
votes
1answer
98 views

Expression for sum of $k$-products of $n$ variables

Given $n$ variables there are $n \choose k$ different terms that are the product of $k$ different variables. For example, in the case that $n = 3$, the $k$-products of the variables $x_1, x_2, x_3$, ...
2
votes
2answers
436 views

Simplifying a Product of Summations

I have, for a fixed and positive even integer $n$, the following product of summations: $\left ( \sum_{i = n-1}^{n-1}i \right )\cdot \left ( \sum_{i = n-3}^{n-1} i \right )\cdot \left ( \sum_{i = ...
4
votes
4answers
98 views

Big Greeks and commutation

Does a sum or product symbol, $\Sigma$ or $\Pi$, imply an ordering? Clearly if $\mathbf{x}_i$ is a matrix then: $$\prod_{i=0}^{n} \mathbf{x}_i$$ depends on the order of the multiplication. But, ...
3
votes
2answers
74 views

How to define this pattern as $f(n)$

Given a binary table with n bits as follows: $$\begin{array}{cccc|l} 2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ ...
3
votes
2answers
96 views

Summation and Product Bounds

If I have a sum or product whose upper index is less than its start index, how is this interpreted? For example: $$\sum_{k=2}^0a_k,\qquad \prod_{k=3}^1b_k$$ I want to say that they are equal to the ...
0
votes
1answer
45 views

Showing an indentity with a cyclic sum

let $z_1,z_2,..,z_n$ be non equal complex numbers for any $n\geqslant2$, for any $k\in \mathbb{N}$ $$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j \ne i}}^{ n }{ ...
1
vote
0answers
75 views

Pi identity with sum and product

Please prove this identity $$\sum_{ n=1 }^{\infty }\left({\left(-1\right) }^{ n }\frac{\prod_{ j=1 }^{ n }{\left(\frac{ 3 }{ 2 }-j\right) }}{\left( 2n+1\right)\left( n!\right) }\right) =\frac{\pi }{ ...
1
vote
2answers
252 views

Maximize the product of linear functions

Suppose $f(x,y) = \prod_{i=1}^n (a_ix+b_iy)$ where $n$ is a constant larger than 500, and $a_i>0$, $b_i>0$ are known coefficient. There is only one global maximum. What's the most efficient ...
1
vote
0answers
79 views

Bounding the product of a sequence

I am trying to find an upper bound for the following sequence: $$(1-p_1)(1-(p_1+p_2))\cdots(1-(p_1+\cdots+p_n))$$ with $n$ groups to multiply. I have written it like this: $$\prod_{i=1}^n \left({1 ...
4
votes
1answer
147 views

Operators - sums, products, exponents, etc.

$(x + x + \cdots + x)$, where $x$ added $n$ times can be written as $x * n$. $(x * x * \cdots * x)$, where $x$ multiplied $n$ times can be written as $x ^ n$. Is there an operator, such that if ...
12
votes
2answers
421 views

proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$

i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows: $$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...