All Questions
299
questions
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Sum of products of K numbers taken from N numbers in closed form
Let's say i have 5 numbers, $A,B,C,D,E$. I want to know the sum of all the possible products of some or all of these numbers each taken at most once.
Instead of a lot of multiplications and additions ...
- 235
0
votes
1
answer
141
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Building matrix expressions for product of sum, isolating vector of constants
This identity to build the matrix expression for the expression below is pretty straightforward:
$$
\left.\sum\limits_{j=1}^M \left( a_j \cdot f_{i,j} \right) \;\right|_{i=1}^N = \left[\begin{array}{}...
- 121
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111
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Summation verification
I have a particular polynomial
$$ 1-10x+35x^2-50x^3 $$
Which can be written nicely as
$$1-(1+2+3+4)x+(1\cdot2+1\cdot3+1\cdot4+2\cdot3+2\cdot4+3\cdot4)x^2$$
$$+(1\cdot2\cdot3+1\cdot2\cdot4+1\cdot3\...
- 8,837
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0
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533
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Simplify the product of two sums
How can I simplify the following product of two sums:
$$
\biggl(\, \sum ^{n}_{k=0}a_{k}\biggr) \biggl(\, \sum ^{n}_{k=0}\dfrac {1}{a_{k}}\biggr)
$$
user188861
0
votes
2
answers
167
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Minimizing sum of products
Consider a total of $d$ items, $\{I_1, I_2, \cdots, I_d \}$, each having a weight $w_i$, and a total of $m$ bins, $\{B_1, B_2, \cdots, B_m\}$. We would like to distribute the items into the bins such ...
- 1
0
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1
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136
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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 $\mathbb C(A)$.
$$\begin{...
- 35.9k
0
votes
1
answer
394
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Summation in 104 Number Theory problems
There's a paragraph of 104 Number Theory problems, on page $9$ that says:
From the formula $\prod_{i=1}^\infty\frac{p_i}{p_i-1} = \infty ,$
using the inequality $1+t \le e^t$, $t \in \mathbb{R}$ we ...
0
votes
1
answer
658
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Multiplying Sigmas(sums)
I would be grateful if someone please rewrite or expand this please.
I have problem multiplying two sigmas ($\sum $)
$$
(d(n)-\sum_{k=-\infty}^{\infty} h_k x(n-k)) \times (d(n)-\sum_{l=-\infty}^{\...
- 1
0
votes
1
answer
75
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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 ...
- 125
0
votes
1
answer
168
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A closed form for $\sum_{i=1}^{n} \prod_{k=1}^{i+2} (3k+2)$
I need to calculate the following expression. Is there any explanation to convert this expression into normal expression without those letters for sum and the product? Just normal expression.
$$ Z ...
-1
votes
1
answer
72
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Showing $(n+m)^{\underline{k}}=\sum_{v=0}^\infty{k\choose v}\cdot{(m)^{\underline{k-v}}}\cdot(n)^{\underline{v}}$ for falling factorials [closed]
Falling and rising factorials are defined as
$$
\color{blue}{n^{\underline{k}}}=\,\,\color{blue}{\prod_{j=0}^{k-1}(n-j)} \qquad\qquad
\color{blue}{n^{\overline{k}}}=\,\,\color{blue}{\prod_{j=0}^{k-1}(...
- 19
-1
votes
2
answers
133
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$\sum_{i=0}^{n}{\prod_{j=0\\j\not=i}^{n}\frac{x-x_j}{x_i-x_j}}=1$ [duplicate]
can you help me to show this identy: $$\sum_{i=0}^{n}{\prod_{j=0\\j\not=i}^{n}\frac{x-x_j}{x_i-x_j}}=1,$$ I try to expand it but I only have that is equal to $$\frac{n\prod_{j=0}^{n}(x-x_j)}{\sum_{i=0}...
- 451
-1
votes
2
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149
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How can I prove this formula: $\sum_{a_1+a_2+\cdots +a_m\le n}\prod_{i=1}^ma_i=\binom{n+m}{2m}$? [closed]
Take $a_1,a_2,\cdots,a_m$ balls sequentially from a box of n balls, and you don't have to take all of them. It is guaranteed that $1\le m\le n$.
Denote the product of $a_1,a_2,\cdots ,a_m$ as $\prod_{...
-3
votes
1
answer
87
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Is there a formula for a summation divided by a product of its terms?
$$\frac{\sum_{i=1}^{n}x_{i}}{\prod_{i=1}^{n}x_{i}}= \frac{1}{x_{2}x_{3}x_{4}...}+\frac{1}{x_{1}x_{3}x_{4}}+\frac{1}{x_{1}x_{2}x_{4}}...$$
There is a very clear pattern that each consecutive result ...
- 173