8
votes
Accepted
Power series where the number $e$ is a root
You can use a series for $W(e^{1+z})$ where $W$ is the Lambert W function. See wikipedia (https://en.wikipedia.org/wiki/Lagrange_inversion_theorem):
$W(e^{1+z}) = 1 + \frac{z}{2} + \frac{z^2}{16} + \...
- 2,329
5
votes
Power series where the number $e$ is a root
Assume $a_0=1/2$ and $a_k\neq 0,$ $0\le k\le n-1,$ are chosen so that $0<S_k<{1\over (k+1)!}$ for $0\le k\le n-1.$ There exists a nonzero negative rational $a_{n}$ satisfying $$0<S_{n}:=S_{n-...
- 33.9k
4
votes
Power series where the number $e$ is a root
Let $\alpha > 1$ be any real umber. Then by invoking the density of the rational numbers, we can choose $a_1, a_2, \ldots \in \mathbb{Q}_{>0}$ such that
$$ 1 - \frac{1}{n!} \leq \sum_{k=1}^{n} ...
- 170k
3
votes
Accepted
Can the elements of this recursive sequence be calculated individually?
All numbers in these sequences are even, so let's divide them by $2$. Here's how you get $\frac12\cdot$ sequence $k$: List the numbers $1,\ldots,2^k-1$ in binary as a $k$-digit number (with leading ...
- 1,548
3
votes
Power series where the number $e$ is a root
Define a sequence $a_0,a_1,a_2,...$ of rational numbers recursively by
\begin{align*}
a_0&\,=\;1
\\[4pt]
a_{n+1}&\,=\;
\begin{cases}
-\frac{1}{3^{n+1}}&\text{if}\;s_n > 0\\[3pt]
\;\;\;\...
- 59.2k
2
votes
Can the elements of this recursive sequence be calculated individually?
Your sequences may be 'extracted' from the sequence OEIS A030109 :
The trick for the sequence $m$ is to reverse the bits of $2^m-1$ consecutive numbers (subract $1$ and divide by $2$) starting at $2^{...
- 43.5k
1
vote
Optimal strategy for uniform distribution probability game
The problem can be formulated as follows: Let $(X_t)_{t \geq 0}$ be a sequence of iid random variables, each is uniformly distributed on $[0,N]$. Adam is the guy who tries to maximize the expected ...
- 1,033
1
vote
Accepted
Telescopic summation for AGP: $R_n=\sum_{k=1}^n k r^{k-1}$
It is easier to use the previous result for the geometric partial sum. We expand
$$kr^{k-1} = \frac{kr^{k-1} - kr^k}{1-r}$$
This is creates a telescopic behaviour, because consecutive terms cancel to ...
- 356
1
vote
Infinite summation formula for modified Bessel functions of first kind
Generated from Mathematica:
$$\int_0^{\frac{\pi }{2}} t I_0(2 \kappa \cos (t)) \, dt=\\\sum _{n=0}^{\infty } \frac{\, _2F_3\left(\frac{1}{2}+n,1+n;1,\frac{3}{2}+n,\frac{3}{2}+n;\kappa ^2\right)}{(1+2 ...
- 4,492
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