All Questions
299
questions
29
votes
1
answer
60k
views
Can we express sum of products as product of sums?
I've got an expression which is sum of products like:
$$a_1 a_2 + b_1 b_2 + c_1 c_2 + \cdots,$$
but the real problem I'm solving could be easily solved if I could convert this expression into ...
- 453
28
votes
2
answers
15k
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 ...
- 943
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
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:
$$\sum\limits_{i = 1}^{n } {\prod\limits_{\substack{j = 1\\j \ne i}}^{n } {\frac{{x_i }}{{x_i - x_j }}} } ...
- 173
12
votes
1
answer
588
views
New Year Maths 2015
In the spirit of the festive period and in appreciation of the encouraging response to my Xmas Combinatorics 2014 problem posted recently, here's one for the New Year!
Express the following as a ...
- 22.8k
12
votes
1
answer
885
views
Showing $\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n b_i \right)^\frac1{n}\le\sum\limits^N_{n=1}\left(\prod\limits_{i=1}^n a_i \right)^\frac1{n}$?
If $a_1\ge a_2 \ge a_3 \ldots $ and if $b_1,b_2,b_3\ldots$ is any rearrangement of the sequence $a_1,a_2,a_3\ldots$ then for each $N=1,2,3\ldots$ one has
$$\sum^N_{n=1}\left(\prod_{i=1}^n b_i \right)^...
- 2,048
11
votes
2
answers
387
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 ...
- 108k
11
votes
0
answers
241
views
Are $(2,28)$ and $(5,3207)$ the only solutions $(m,n)\in\mathbb{N}^2$?
I noticed something as I was playing around with prime numbers. By denoting $p_i$ the $i^{\text{th}}$ prime number, I discovered the following:
$$
\begin{align}\prod_{i=1}^2\left(p_i^{ \ 2}+i\right)&...
- 9,487
10
votes
1
answer
717
views
Product of Sines and Sums of Squares of Tangents
There is a nice formula for products of cosines, found by multiplying by the complementary products of sines and using the double angle sine formula (as I asked in my question here): $$\prod_{k=1}^n \...
- 8,945
10
votes
4
answers
370
views
Geometry problem boils down to finding a closed form for $\sum_{n=1}^{k}{\arctan{\left(\frac{1}{n}\right)}}$
I was solving the following problem:
"Find $\angle A + \angle B + \angle C$ in the figure below, assuming the three shapes are squares."
And I found a beautiful one-liner using complex numbers:
$(1+...
- 596
10
votes
1
answer
508
views
Why is this sum equal to $0$?
While solving a differential equation problem involving power series, I stumbled upon a sum (below) that seemed to be always equal to $0$, for any positive integer $s$.
$$
\sum_{k=0}^s \left( \frac{ \...
- 365
10
votes
1
answer
476
views
If integration is a continuous analog of summation (Addition), what is the continuous analog of multiplication (Product)?
One definition of integration over a continuous interval [a,b] into n subintervals with equal width $\Delta x$, and from each interval choose a point $x_i^*$. Then the definite integral of $f(x)$ ...
- 101
9
votes
3
answers
487
views
Question about Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$
For a freshman calculus project, I used Euler's approach to find $\sum_{n=1}^\infty\frac1{n^2}=\frac{\pi^2}6$, and noted from Wikipedia's explanation that the infinite product representation of $\frac{...
- 265
8
votes
4
answers
21k
views
Change from product to sum
We know that : $$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
8
votes
2
answers
373
views
Showing an indentity with a cyclic sum
Let $n\geqslant2$, and $k\in \mathbb{N}$
Let $z_1,z_2,..,z_n$ be distinct complex numbers
Prove that
$$ \sum_{i=1}^{n}\frac {{z}_{i}^{n-1+k}} { \prod \limits_{\substack{j = 1\\j \ne i}}^{ n }{ (z_i-...
- 35.9k
8
votes
1
answer
467
views
Identity in Number Theory Paper
In this paper by Jerry Hu, he defines the function
$$f_{s,k,i}\left(u\right)=\prod_{p\mid u}
\left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose m}\left(p-...
- 2,155
8
votes
0
answers
258
views
Help me to get deeper understanding of Euler's proof of his Arithmetical Theorem
With distinct numbers $a_1, a_2, \ldots, a_n$, let's denote the products of the differences of each of these numbers with the each of the rest of them by the following principle:
\begin{align}
(...
- 848
7
votes
1
answer
365
views
How to prove the following discovery of Euler?
There exists a series of formulas.
\begin{align*}
\ & \dfrac{1}{(a-b)(a-c)}+\dfrac{1}{(b-a)(b-c)}+\dfrac{1}{(c-a)(c-b)} = 0, \\
\ & \dfrac{a}{(a-b)(a-c)}+\dfrac{b}{(b-a)(b-c)}+\dfrac{c}...
- 848
7
votes
3
answers
1k
views
Summation series ($\Sigma$) is to Integral ($\int$)... as Product series ($\Pi$) is to ??
If a Summation series ($\Sigma$) is to an Integral ($\int$)... is there a corresponding concept for a Product series ($\Pi$)?
Summation series ($\Sigma$) is to Integral ($\int$)... as Product series (...
7
votes
2
answers
200
views
Elementary proof of "generalized reverse Bernoulli inequality"
I've stumbled upon the following exercise in an early chapter of an analysis textbook:
Let $a_n$ be a finite, nonnegative sequence such that $\sum_{i=0}^n a_i\le 1$. Prove $$ \prod_{i=1}^n (1 + a_i) \...
- 878
6
votes
2
answers
359
views
Simplify $\prod_{k=1}^5\tan\frac{k\pi}{11}$ and $\sum_{k=1}^5\tan^2\frac{k\pi}{11}$
My question is:
If $\tan\frac{\pi}{11}\cdot \tan\frac{2\pi}{11}\cdot \tan\frac{3\pi}{11}\cdot \tan\frac{4\pi}{11}\cdot \tan\frac{5\pi}{11} = X$ and $\tan^2\frac{\pi}{11}+\tan^2\frac{2\pi}{11}+\tan^2\...
- 2,586
6
votes
2
answers
144
views
Combinatorics Problem on proving that a particular sum is 0
I'm having some issues with proving that the following sum is $0$ for any value of $n \geq 2$:
$$
\sum_{j=1}^{n} \frac{1}{\prod_{i=1,i\neq j}^{n}(a_{j}-a_{i})}
$$
where the $a_i$ are non-zero and ...
- 63
6
votes
2
answers
199
views
Calculate $\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$
Calculate $$\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2$$
I can only bound it as follows:
$$\binom{n}{i}<\left(\dfrac{n\cdot e}{k}\right)^k$$
$$\sum_{i = 0}^{n}\ln\binom{n}{i}\Big/n^2<\dfrac{1}{n}\...
- 884
6
votes
0
answers
97
views
Prove that if $x_{1}x_{2}...x_{n}=1$ then $\frac{1}{1+x_{1}+x_{1}x_{2}}+...+\frac{1}{1+x_{n-1}+x_{n-1}x_{n}}+\frac{1}{1 +x_{n}+x_{n}x_{1}}\ge 1$ [duplicate]
Prove that if $x_{1}x_{2}...x_{n}=1$ then $\frac{1}{1+x_{1}+x_{1}x_{2}}+...+\frac{1}{1+x_{n-1}+x_{n-1}x_{n}}+\frac{1}{1 +x_{n}+x_{n}x_{1}}\ge 1$.
$x_{1},x_{2},...,x_{n}$ are positive real numbers, and ...
- 288
5
votes
4
answers
485
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, ...
- 1,489
5
votes
3
answers
3k
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?
- 2,883
5
votes
2
answers
151
views
An equality between a product and a combinatorial sum
I'm trying to prove the following identity (of which I numerically verified the truth) :
$$\text{For every $n\in\mathbb{N}^*$ and $\alpha \in \mathbb{R}\setminus\lbrace-2k\text{ }|\text{ }k\in\mathbb{...
- 1,984
5
votes
3
answers
195
views
Prove that $\frac{\sqrt[n]{\prod_{k = 1}^nx_n}}{m} \ge n - 1$ where $\sum_{k = 1}^n\frac{1}{x_k + m} = \frac{1}{m}$.
Given positives $x_1, x_2, \cdots, x_{n - 1}, x_n$ such that $$\large \sum_{k = 1}^n\frac{1}{x_k + m} = \frac{1}{m}$$. Prove that $$\large \frac{\displaystyle \sqrt[n]{\prod_{k = 1}^nx_n}}{m} \ge n - ...
- 4,246
5
votes
1
answer
1k
views
Sum operator precedence
I'm trying to read some simple equations and in order to interpret them in the right way I need to know $\sum$ and $\prod $ operator range/precedence.
$$ \sum p(s, a) +\gamma $$
is equal to $\sum(p(...
- 177
5
votes
1
answer
225
views
Proving $(1+\frac 1n)^{n} = 1 + \sum_{k=1}^n({\frac 1{k!}\prod_{r=0}^{k-1}(1-\frac rn))}$ using the binomial theorem
$$\left(1+\frac 1n\right)^{n} = 1 + \sum\limits_{k=1}^n \left\{\frac 1{k!}\prod_{r=0}^{k-1}\left(1-\frac rn\right)\right\}$$
this exercise is taken from Apostol's Calculus I (page 45) and it's ...
- 135
5
votes
1
answer
320
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 $x^{...
- 711
5
votes
2
answers
169
views
Prove that $\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$
For arbitrary $x$ and $1\leqslant m\leqslant n$, prove the following:
$$\sum_{k=1}^n\frac{\prod_{1\leq r\leq n, r\neq m}(x+k-r)}{\prod_{1\leq r\leq n, r\neq k}(k-r)}=1$$
I'm looking for a proof that ...
- 848
5
votes
1
answer
468
views
Average weighted by inverse distance to median equal to median?
Problem Statement
I have a set of $N$ ordered elements such that $x = \{x_1, x_2, ..., x_q, x_p, ..., x_N\}$ where $x_q \le x_m \le x_p$ and $x_m$ is the median of the set $x$. I define a particular ...
- 205
5
votes
1
answer
156
views
Permutations of Independent Probabilities
Problem:
Say that I have a list of $n$ tasks to complete. Each of the tasks have independent probabilities $p_1, p_2, ..., p_n$ of completing that task.
There is a particular task on the list that I ...
- 51
4
votes
2
answers
272
views
Sum of positive elements divided by their "weighted" product - inequality
I have following expression,
$$ \frac{\sum_{i=1}^n x_i}{\prod_{i=1}^nx_i^{p_i}} $$
where $p_i$s satisfy $\sum p_i = 1$ and $p_i \in [0,1]$ and $x_i\geq0$, $\forall i \in 1\dots n$.
I think that ...
- 221
4
votes
3
answers
867
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 :
$$a=\sum_{r=1}^8 \tan^2\left(\frac{r\pi}{17}\right) \qquad\qquad b=\prod_{r=1}^8 \tan^2\left(\...
- 183
4
votes
2
answers
191
views
sum of an infinite series $\sum_{k=1}^\infty \left( \prod_{m=1}^k\frac{1}{1+m\gamma}\right) $
I am trying to find a closed form expression of
$$
\sum_{k=1}^\infty \left( \prod_{m=1}^k\frac{1}{1+m\gamma}\right)
$$
where $\gamma>1$.
I've been trying this for a long time. Is there an easy way ...
- 143
4
votes
2
answers
590
views
Coefficients of $(x-1)(x-2)\cdots(x-k)$
I'm interested in the coefficients of $x$ in the expansion of,
$$ (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),$$
Where $k$ is an integer. In particular I am ...
- 12.4k
4
votes
2
answers
3k
views
Simplifying a Product of Summations
I have, for a fixed and positive even integer $n$, the following product of summations:
$\left ( \sum_{i = n-1}^{n-1}i \right )\cdot \left ( \sum_{i = n-3}^{n-1} i \right )\cdot \left ( \sum_{i = n-...
4
votes
4
answers
305
views
Summation of reciprocal products
When studying summation of reciprocal products I found some interesting patterns.
$$\sum_{k=1}^{N} \frac{1}{k\cdot(k+1)}=\frac{1}{1\cdot1!}-\frac{1}{1\cdot(N+1)}$$
$$\sum_{k=1}^{N} \frac{1}{k\cdot(k+1)...
- 795
4
votes
3
answers
104
views
$Q\le \prod \frac{5+2x}{1+x}\le P$ find $P,Q$
if $x,y,z,$ are positives and $x+y+z=1$ and $$Q\le \prod_{cyc} \frac{5+2x}{1+x}\le P$$ find maximum value of $Q$ and minimum value of $P$
This is actually a question made up myself ,so i don,t know ...
- 11.9k
4
votes
1
answer
304
views
Proving $\sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq k}(x_k-x_j)}=\sum_{k=0}^nx_k$
In Problems from the book by Andreescu, there's the following problem :
Let $x_0,\ldots,x_n$ be distinct complex numbers.
Prove $\displaystyle \sum_{k=0}^n\dfrac{x_k^{n+1}}{\prod_{j\neq k}(...
- 35.9k
4
votes
3
answers
159
views
Product of sums which equal to sum of product
We can be sure that
$$\left(\sum\limits_{k=0}^{n}\frac{1}{k+1}\right)\left(\sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{k+1}\right)= \sum\limits_{k=0}^{n}\binom{n}{k}\frac{(-1)^k}{(k+1)^2}$$
Is ...
- 1,475
4
votes
2
answers
637
views
Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$
I want to prove
$$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$
if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$
I have no idea where to start, any advice would be greatly appreciated!
- 143
4
votes
2
answers
136
views
Show that $k^a=\sum_{m=1}^b\left ( c_m^a\prod_{n\neq m} \frac{k-c_n}{c_m-c_n} \right ).$
I used the following result in another post without providing proof (because I couldn't prove it):
$$k^a=\sum_{m=1}^b\left ( c_m^a\prod_{n\neq m} \frac{k-c_n}{c_m-c_n} \right ),$$
where $a$ and $b$ ...
- 2,550
4
votes
1
answer
7k
views
Can Pi prod be expressed using Sigma Notation?
Can $\prod(x)$ be expressed in terms of $\sum (x)$?
- 6,834
4
votes
2
answers
82
views
How to define this pattern as $f(n)$
Given a binary table with n bits as follows:
$$\begin{array}{cccc|l}
2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ &0&1&0&...
- 265
4
votes
1
answer
88
views
Formulating an alternating sum of product combinations
Consider some list $A=(a_1,a_2,\cdots,a_n)$. I'd like to find a closed form for the following operation.
$$f(A)=\sum_{k=1}^n(-1)^{k-1}s_k= s_1-s_2+\cdots(-1)^{n-1}s_n.$$
Where $s_k$ is the sum of all ...
- 4,472
4
votes
1
answer
115
views
How can I evaluate the below mentioned series without using a computation software?
I have been trying to evaluate $\displaystyle\sum_{m=0}^{2^{2^5}-1}\frac{2}{\prod_{n=1}^5\bigl((m+2)^{\frac{2}{n}}+(m)^{\frac{2}{n}}\bigr)}$ for quite a long time. I tried various approaches but ...
- 121
4
votes
1
answer
1k
views
Sum and Product Puzzle and Prime Factors
Suppose we have two number $X$ and $Y,$ such that $1 < X < Y < 100,$ and $X + Y ≤ 100.$ Sue is given $S = X + Y$ and Pete is given $P = XY.$ They then have the following conversation:
Pete: '...
- 143