All Questions
29
questions
0
votes
0
answers
44
views
Inequality with Products and Sums
I need help to find a proof for the following inquality.
Assuming that $ 0 \leq c_i \leq 1 $ and $ 0 \leq d_i \leq 1 $, show that
$$
\prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
- 778
2
votes
1
answer
128
views
Let $A_{k}=\{0,... ,n\}\setminus\{k\}.$ How to prove $\sum_{k=0}^{n}\left[(-1)^{k+1}\prod_{\substack{i,j\in A_{k}\\i<j}}(a_{i}-a_{j})\right]=0$?
Let $A_{k}=\{0,1,\ldots,n\}\setminus\{k\}$ for each $k=0,1,\ldots ,n$.
I think that the following equality is true for all $n\in\mathbb{N}, n\geq 2$ :
\begin{align}
\sum_{k=0}^{n}\left[(-1)^{k+1}\...
- 23
2
votes
1
answer
47
views
Double Sum to Product Derivation
The function after the double-sigma sign can be separated into the
product of two terms, the first of which does not depend on $s$ and
the second of which does not depend on $r$. Source
Is the ...
- 1,818
1
vote
1
answer
74
views
Simplifying $\sum\limits^n_{i=1}\bigl(\prod\limits^n_{j=1}\bigl(a_j\bigl\lfloor\frac{x_j-x_i}{|x_j-x_i|+1} \bigr\rfloor+1\bigr)\bigr)b_i$
I have an indexed finite set of elements $X = \{x_1,x_2,x_3,...,x_n\}$, where $x_i \in \mathbb{R}$. And a corresponding indexed finite set $A = \{a_1,a_2,a_3,...,a_n\}$, where $a_i \in [0,1]$ and a ...
- 197
2
votes
2
answers
119
views
Product of $n$ terms of sequence where the $n^{th}$ term is of the form $(x^{a^n}+1)$
While practicing from a book I found a product in the form $$(x^{a^1}+1)\cdot(x^{a^2}+1)\cdot(x^{a^3}+1)\cdot(x^{a^4}+1)$$ and was immediately curious if I could a formula to solve the product for $n$ ...
- 121
2
votes
3
answers
92
views
Expansion of $\prod_{j=1}^{n} \left( \sum_{i = 1}^{m} x_{i, j} \right)$
I would like to know if there is a sum-of-products expansion for the following product-of-sums. A special case is given here for the difference of two entries.
$$\prod_{j=1}^{n} \left( \sum_{i= 1}^{m} ...
- 1,878
1
vote
1
answer
194
views
I wish to solve exactly this formula involving sums and products
I was solving a physics exercise and I encountered this formula:
$$\left< n_l \right>=\left[1+\sum_{k\neq l} \left(e^{bN(l-k)}\frac{\prod_{j\neq l} (1-e^{b(l-j)})}{\prod_{j\neq k} (1-e^{b(k-j)})}...
- 56
1
vote
1
answer
45
views
Sum and product problem
How can I find the result of:
$\sum\limits_{i=1}^{n}\prod\limits_{j=1}^{2} ij$
I know that $\prod\limits_{j=1}^{2} ij = 2i^2$, so I should simply do the summation as $\sum\limits_{i=1}^n2i^2$?
4
votes
3
answers
860
views
Finding $\frac{\sum_{r=1}^8 \tan^2(r\pi/17)}{\prod_{r=1}^8 \tan^2(r\pi/17)}$
I have tried to wrap my head around this for some time now, and quite frankly I am stuck.
Given is that :
$$a=\sum_{r=1}^8 \tan^2\left(\frac{r\pi}{17}\right) \qquad\qquad b=\prod_{r=1}^8 \tan^2\left(\...
- 183
3
votes
1
answer
92
views
Proving that, if $f(k)=\prod_{i=1}^ka_i+\sum_{b=1}^{k-1}(1-a_{k-b})\prod_{i=1}^ba_{k-b+i}$, then $f(k+1)=f(k)\cdot a_{k+1}+(1 - a_k)a_{k+1}$
Given that $$f(k) = \prod_{i=1}^k a_i + \sum_{b=1}^{k-1} (1-a_{k-b}) \prod_{i=1}^b a_{k-b+i}$$ for all $k$ where $(a_1, a_2, a_3, \ldots )$ are random constants, prove that: $$f(k+1) = f(k) \cdot a_{k+...
user851578
2
votes
3
answers
68
views
Is there an easy way of seeing that this sum is $0$?
Let $a,b,c,d$ be real positive numbers. Is there an easy way of seeing that
$$(c-d)(b-d)(b-c)-(c-d)(a-d)(a-c)+(b-d)(a-d)(a-b)-(b-c)(a-c)(a-b)=0 $$
without doing the whole multiplication ?
- 2,294
0
votes
3
answers
2k
views
If $\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right)= \prod_{r = 10}^{99}\log _r(r+1)$, then find $n$.
If \begin{align}\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right) = \prod_{r = 10}^{99}\log _r(r+1).\end{align}
then find $n$.
I found this question in my 12th grade textbook and I just can't wrap ...
- 125
2
votes
2
answers
146
views
Find the value of $a_0a_1a_2\cdots a_n\left(\frac{1}{a_0}+\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}\right)$
Given that the sequence $\left\{a_n\right\}$ satisfies $a_0 \ne 0,1$ and $$a_{n+1}=1-a_n(1-a_n)$$ $$a_1=1-a_0$$
Find the value of $$a_0a_1a_2\cdots a_n\left(\frac{1}{a_0}+\frac{1}{a_1}+\frac{1}{a_2}+...
- 13.2k
0
votes
1
answer
40
views
Two sums out of a product
Problem
Assuming $\theta \in [0, 1], y_i \in \{0, 1\}$,
I'm having trouble deriving the expression on the right from the one on the left:
$$
\begin{align}
\prod_i \theta^{y_i}(1-\theta)^{(1-y_i)} ~~~...
- 2,055
0
votes
1
answer
180
views
The Lagrange Interpolation formula – Spivak's Calculus Ch 3 Problem 7(b)
The problem: Now find a polynomial function $f$ of degree $n - 1$ such that $f(x_i) = a_i$, where $a, \ldots, a_n$ are given numbers.
I found that this question had been asked before, but I did not ...
- 87