All Questions
298
questions
2
votes
0
answers
20
views
Are there any other sequences of functions besides $\prod_{j=0}^{k-1}(x+j) $ for which both their sum and the sum of their reciprocals telescope?
If
$p_k(x)
=\prod_{j=0}^{k-1}(x+j)
$,
then both
$\sum_{j=a}^b p_k(x+j)
$
and
$\sum_{j=a}^b \dfrac1{p_k(x+j)}
$
can be written as
telescoping sums
so we get
$$\sum_{j=a}^b p_k(x+j)
=\dfrac1{k+1}\left(...
- 108k
0
votes
0
answers
44
views
Induction proof for product of $a^x$ is less than or equal to the sum of $x\times a$
So this type of problem has me stuck in proving some relation. I assumed to use induction but I am stuck at a certain step and cannot understand if there is a trick or perhaps my idea is just wrong:
...
- 33
3
votes
0
answers
68
views
Evaluating a summation of product
Show that for any integer $k>1$
$$
\sum_{\substack{i_j \in \{0, 1\} \forall j < k, i_k = 0}} \prod_{j = 1}^{k} \left(i_j + (-1)^{i_j} \frac{a+ (j - 1) c - c \sum_{\lambda = 0}^{j - 1} i_\lambda}{...
- 51
3
votes
0
answers
38
views
Convoluted notation involving a double sum and a product $\sum_{m\geq0}\sum_{1\leq...<|x|;1\leq...<|y|}$ where $x$ and $y$ are sequences
Theorem 4 The discretized signature kernel over $k$,
$$
\mathrm{k}^{+}: X^{+} \times X^{+} \rightarrow \mathbb{R}, \quad \mathrm{k}^{+}(x, y)=\left\langle\mathrm{S}^{+}\left(\mathrm{k}_x\right), \...
0
votes
1
answer
173
views
Simplifying tricky sum of products
Could someone please clearly write out how we get from this expression
$$\log\left[\sum_{\mathbf Z}\left(\prod_{n=1}^N\prod_{m=1}^M\pi_m^{\mathbf{1}(z_n=m)}\mathcal{N}\left(\mathbf x_n;\mathbf{\mu}_m,\...
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
$$
\prod_{i=1}^N (c_i + d_i - c_i d_i) \geq \prod_{i=1}^N c_i + \prod_{...
- 778
0
votes
2
answers
199
views
how to calculate $\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$
Question: how to calculate $$\sum\limits_{k=1}^{+\infty }{\arctan \frac{1}{1+k^{2}}}$$
My attempt
Let
$\arctan \theta =\frac{i}{2}\ln \left( \frac{i+\theta }{i-\theta } \right)$
$$S=\sum\limits_{k=1}^{...
1
vote
1
answer
52
views
Sums and Products over sets of sets
Say I have a set of sets eg. $A:=\{\{1,2\},\{2,3\}\}$. Does this formula
$$\sum_\limits{A_i \in A}\prod_{j\in A_i}x_j$$
yield $x_1x_2+x_2x_3$?
Or does it even make sense to define a sum over a set of ...
- 11
2
votes
1
answer
65
views
Solutions $X_{k}$ to the equation $\sum_{k=0}^N X_{k} (-2^k)^n = 1$
In the proof of the lemma in this paper, the author makes the following claim: Fix integer $N$ and let $n \leq N$. Then the solutions $X_0,\ldots,X_N$ to the equations
$$\sum_{k=0}^N X_{k} (-2^k)^n = ...
- 199
0
votes
0
answers
23
views
A hypothesis For $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$
(Hypothesis) For integer $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$, then the following identity, whether or not to be established:
$$
\prod_{j=1}^m \text{Li}_{l_j}\left[ x_j \right] = \sum_{k=0}^...
user1235604
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
0
votes
0
answers
59
views
Summation with inner products: properties and rearrangement
OPTION 1.
I have this expression,
$$\sum \limits_{l=1}^{L}a_l^2 + \sum \limits_{l<j}^{L}a_l^2 \frac{a_j}{a_l}$$
and I would like to take out $$\sum \limits_{l=1}^{L}a_l^2$$, as to obtain something ...
- 181
1
vote
1
answer
61
views
Can you interchange sums and products(in Bishop’s book, pattern recognition and machine learning)?
According to this stackexchange thread,
cannot swap sums and products
, you cannot interchange sums and products.
$$ \Sigma\Pi x_{i,j} \ne \Pi\Sigma x_{i,j} $$
However, I found in the book, equations ...
- 13
2
votes
1
answer
84
views
Proving an inequality consisting of sums and products
I have a tricky inequality (related to some previous ones that I posted) which I am yet again stuck trying to solve. I have confirmed that it is true through simulation (at least up until overflow ...
- 183