All Questions
29
questions
0
votes
0
answers
45
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
$$
\prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
- 778
2
votes
1
answer
129
views
Let $A_{k}=\{0,... ,n\}\setminus\{k\}.$ How to prove $\sum_{k=0}^{n}\left[(-1)^{k+1}\prod_{\substack{i,j\in A_{k}\\i<j}}(a_{i}-a_{j})\right]=0$?
Let $A_{k}=\{0,1,\ldots,n\}\setminus\{k\}$ for each $k=0,1,\ldots ,n$.
I think that the following equality is true for all $n\in\mathbb{N}, n\geq 2$ :
\begin{align}
\sum_{k=0}^{n}\left[(-1)^{k+1}\...
- 23
2
votes
1
answer
47
views
Double Sum to Product Derivation
The function after the double-sigma sign can be separated into the
product of two terms, the first of which does not depend on $s$ and
the second of which does not depend on $r$. Source
Is the ...
- 1,834
1
vote
1
answer
74
views
Simplifying $\sum\limits^n_{i=1}\bigl(\prod\limits^n_{j=1}\bigl(a_j\bigl\lfloor\frac{x_j-x_i}{|x_j-x_i|+1} \bigr\rfloor+1\bigr)\bigr)b_i$
I have an indexed finite set of elements $X = \{x_1,x_2,x_3,...,x_n\}$, where $x_i \in \mathbb{R}$. And a corresponding indexed finite set $A = \{a_1,a_2,a_3,...,a_n\}$, where $a_i \in [0,1]$ and a ...
- 197
2
votes
2
answers
120
views
Product of $n$ terms of sequence where the $n^{th}$ term is of the form $(x^{a^n}+1)$
While practicing from a book I found a product in the form $$(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} ...
- 1,876
1
vote
1
answer
196
views
I wish to solve exactly this formula involving sums and products
I was solving a physics exercise and I encountered this formula:
$$\left< n_l \right>=\left[1+\sum_{k\neq l} \left(e^{bN(l-k)}\frac{\prod_{j\neq l} (1-e^{b(l-j)})}{\prod_{j\neq k} (1-e^{b(k-j)})}...
- 56
1
vote
1
answer
45
views
Sum and product problem
How can I find the result of:
$\sum\limits_{i=1}^{n}\prod\limits_{j=1}^{2} ij$
I know that $\prod\limits_{j=1}^{2} ij = 2i^2$, so I should simply do the summation as $\sum\limits_{i=1}^n2i^2$?
4
votes
3
answers
868
views
Finding $\frac{\sum_{r=1}^8 \tan^2(r\pi/17)}{\prod_{r=1}^8 \tan^2(r\pi/17)}$
I have tried to wrap my head around this for some time now, and quite frankly I am stuck.
Given is that :
$$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 $$
without doing the whole multiplication ?
- 2,294
0
votes
3
answers
2k
views
If $\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right)= \prod_{r = 10}^{99}\log _r(r+1)$, then find $n$.
If \begin{align}\sum_{r=0}^{n-1}\log _2\left(\frac{r+2}{r+1}\right) = \prod_{r = 10}^{99}\log _r(r+1).\end{align}
then find $n$.
I found this question in my 12th grade textbook and I just can't wrap ...
- 125
2
votes
2
answers
147
views
Find the value of $a_0a_1a_2\cdots a_n\left(\frac{1}{a_0}+\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}\right)$
Given that the sequence $\left\{a_n\right\}$ satisfies $a_0 \ne 0,1$ and $$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:
$$
\begin{align}
\prod_i \theta^{y_i}(1-\theta)^{(1-y_i)} ~~~...
- 2,055
0
votes
1
answer
180
views
The Lagrange Interpolation formula – Spivak's Calculus Ch 3 Problem 7(b)
The problem: Now find a polynomial function $f$ of degree $n - 1$ such that $f(x_i) = a_i$, where $a, \ldots, a_n$ are given numbers.
I found that this question had been asked before, but I did not ...
- 87
0
votes
1
answer
54
views
Does $\exp(\sum_i a_i \log(a_i)) = \prod_i a_i^{a_i}$?
I rewrite here the question to avoid visualization problems.
Does the following hold (for $a_i \in \mathbb R$ and $a_i>0$)?
$$e^{\sum_i a_i \log(a_i)}=\prod_i a_i^{a_i}$$
- 779
1
vote
0
answers
48
views
Transforming a long sum of products for efficient computation (based on spanning trees in a complete graph)
Let's say there is a set of $n$ real coefficients: $a_1,...,a_n$. My task is to calculate the value of a rather simple sum of k products: ...
- 239
10
votes
1
answer
717
views
Product of Sines and Sums of Squares of Tangents
There is a nice formula for products of cosines, found by multiplying by the complementary products of sines and using the double angle sine formula (as I asked in my question here): $$\prod_{k=1}^n \...
- 8,945
1
vote
0
answers
35
views
Help with simplification of this expression.
So i have derived this expression and would like to simplify it (i.e find an expression purely in terms of J and v).
$$ J\bigg[1+\sum_{n=1}^{J-1}\prod_{k=1}^{n}\frac{k(1-v)(J-k)}{(k+1)(J-k-1 + vk)}\...
- 61
0
votes
1
answer
22
views
How to expand $\prod_{j=1}^{r}x_{j}^{-1}=\sum_{j=1}^{r}x^{-1} \prod_{j=1,k \neq j}^{r}x_{k}^{-1}$?
I want to know how to expand the
$$\prod_{j=1}^{r}x_{j}^{-1}=\sum_{j=1}^{r}x^{-1} \prod_{j=1,k \neq j}^{r}x_{k}^{-1}$$
where $x=\sum_{j=1}^{r}x_{j}$.
I tried to do this for $r=3$
$$\frac{1}{...
- 517
3
votes
1
answer
64
views
What is condition that the sum of $n$ complex numbers eaquals their product
Let $n\geq2$ and let $\{z_1,\dots,z_n\}$ be a set of complex numbers.
Is there a condition on the $z_i$'s such that
$$\sum_{i=1}^n z_i=\prod_{i=1}^n z_i$$
is identically true?
For $n=2$ the ...
- 1,008
2
votes
1
answer
2k
views
Product of two sums, one finite and one infinite
I'm working on a problem and I'm not sure how to find the product of these two sums:
$\left(\sum_{k=0}^{\infty}\text{something}\right)\left(\sum_{k=n}^{n}\text{something else}\right)$
The "something"...
- 371
1
vote
2
answers
64
views
Verify the Product of a Summation
Can anybody verify that the below equation equals $0$?
$\prod\limits_{k=2}^{10} (\sum\limits_{i=1}^{k-1}(2(i-1)))$
Here is my work, I believe it's correct:
Note: The sequence continues, I just didn'...
user283028
8
votes
4
answers
21k
views
Change from product to sum
We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$
That's how we convert from a product to a sum.
So what happens if we go a little further?
That is : $$\prod\limits_{a}^...
- 2,221
2
votes
1
answer
209
views
On finite sums and products
I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for ...
- 21
2
votes
0
answers
150
views
Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$
For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate
$n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$
So the product goes up to $k$ and I ...
- 69
4
votes
2
answers
82
views
How to define this pattern as $f(n)$
Given a binary table with n bits as follows:
$$\begin{array}{cccc|l}
2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ &0&1&0&...
- 265
17
votes
2
answers
862
views
proof of $\sum\nolimits_{i = 1}^{n } {\prod\nolimits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } = 1$ [duplicate]
i found a equation that holds for any natural number of n and any $x_i \ne x_j$ as follows:
$$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
- 173
26
votes
1
answer
860
views
Is this algebraic identity obvious? $\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1$
If $\lambda_1,\dots,\lambda_n$ are distinct positive real numbers, then
$$\sum_{i=1}^n \prod_{j\neq i} {\lambda_j\over \lambda_j-\lambda_i}=1.$$
This identity follows from a probability calculation ...
user940