All Questions
299
questions
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