All Questions
10
questions
0
votes
1
answer
54
views
Why is $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{2(\lfloor xs\rfloor - sp)^2}{s}\right)}$
I am trying to understand few of the mathematical steps I have encountered in a paper, there are two of them
(a) $\sum_{m=0}^{\lfloor xs\rfloor} 2 \binom{s}{m} p^m (1-p)^{s-m} \leq 2\exp{\left(-\frac{...
- 141
2
votes
0
answers
60
views
How to simplify $e^{-\sum_{i=-k}^k(k-|i|)x_i}$?
Consider the expression given by
$$
\large e^{-\Large\sum_{i=-k}^k(k-|i|)x_i}
$$
Is there a way of simplifying this expression?
For example, provided $\{x_i\}$ is bounded and "smooth" enough ...
- 3,435
0
votes
1
answer
48
views
How to simplify $\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}$
Is it possible to simplify
$$
S(\mathbf{x})=\sum_{k=0}^{n} \frac{e^{\sum_{i=0}^kx_{i}}}{\sum_{i=0}^k x_{i}}
$$
A few observations:
$\sum_{k=0}^{n}\sum_{i=0}^k x_{i}=\sum_{k=0}^{n}(n+1-k)x_k$
$ e^{\...
- 3,435
0
votes
1
answer
40
views
given the sum of a finite sequence of real numbers $x_i$'s, find the $\sum_{i=1}^{N} e^{x_i}$
Let $\sum_{i=1}^{N} x_i $$=$$ 1 $ then what could one say about $\sum_{i=1}^{N} e^{x_i} $$=$$ ? $
- 159
2
votes
2
answers
72
views
Interesting Inequality With Exponents And Base > 1
I had trouble proving the following inequality:
$\beta > 1$
$(\alpha_{1}\beta^{2\alpha_{1}} + \ldots + \alpha_{n}\beta^{2\alpha_{n}})(\beta^{\alpha_{1}} + \ldots + \beta^{\alpha_{n}}) \geq (\...
- 205
1
vote
2
answers
127
views
Given $S_n = \sum \dots$ and $a_n = \sum \dots$ prove that $a_n = S_n + {1\over n\cdot n!}$
I'm trying to solve the following problem:
Let:
$$
\begin{align}
S_n &= 2 + {1\over2!} + {1\over3!} + {1\over 4!} + \dots + {1\over n!} \\
a_n &= 3 - {1\over 1\cdot2\cdot2!} - {1\over 2\cdot3\...
- 5,411
6
votes
4
answers
577
views
Find closed formula by changing order of summation: $\sum_{i=1}^ni3^i$
Working on homework for a probability and computing class, but my ability to work with summations is rusty to say the least, so I suspect this is going to turn out pretty straightforward.
Problem ...
- 73
3
votes
4
answers
186
views
Summation or Integral representation ${e^{2}\above 1.5pt \ln(2)}=10.66015459\ldots$
How can I construct a summation or integral representation of $${e^2\above 1.5pt \ln(2)}.$$ Naively I would write $$\Bigg(\sum_{n=0}^{\infty}{2^n \above 1.5pt n!} \Bigg)\Bigg(\sum_{n=1}^\infty {(-1)^{...
- 3,788
0
votes
0
answers
66
views
Limit in sum and fraction
I'm staring at
$$ f(n) = \sum_{x=1}^n a(n)^{n-x}\\
a(n) = \frac{(1 - \frac{f}{n})y}{y(1 - \frac{f}{n} - \frac{\delta}{n}) + \frac{\delta}{n}}$$
where $f$, $y$, $\delta$, all $ \in (0, 1)$, and $n$ ...
- 1,089
1
vote
2
answers
180
views
Solving $x^{2n} = \frac{1}{2^n}$ for $x$
What is the principle behind solving for a variable that is raised to another variable? I came across this problem doing infinite sums: I had to solve the equation
$$x^{2n} = \frac{1}{2^n}$$
for $...
- 11