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Results tagged with sequences-and-series
Search options questions only
not deleted
user 140356
for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
12
votes
1
answer
954
views
Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$
Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$.
You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does pret …
- 3,098
10
votes
2
answers
929
views
Erroneous Wolfram result for $\sum_{k=1}^\infty (k^3 + a^3)^{-1}$, looking for correct formula
I was trying to get some interesting result for $\zeta(3)$, exploring the following function:
$$W(a) = \sum_{k=1}^\infty \frac{1}{k^3 + a^3}, \mbox{ with } \lim_{a\rightarrow 0} W(a) = \zeta(3).$$
Let …
- 3,098
8
votes
2
answers
340
views
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
Does $x_0=1/3$ lead to periodicity in the logistic map $x_{k+1}=4x_k(1-x_k)$?
I believe it does not, but this is equivalent to proving that $(2\pi)^{-1}\arcsin(\sqrt{1/3})$ is irrational. I am wonderi …
- 3,098
5
votes
3
answers
2k
views
How many digits of $\sqrt{2}$ are known to date?
How many digits of $\sqrt{2}$ are known to date, in base 10 and in base 2? I am trying to produce the largest sequence known to date, and would like to sense if I can do it either alone or with hiring …
- 3,098
5
votes
2
answers
595
views
Sequences similar to $\{n\alpha\}$ that are both equidistributed and truly random-like
See update at the bottom.
Here the brackets represent the fractional part, and $\alpha \in [0, 1]$ is a positive irrational number. It is well known that the sequences $\{n\alpha\}$, $\{n^2\alpha\}$ a …
- 3,098
4
votes
1
answer
607
views
Asymptotics for $\prod(1-\frac{1}{p})$ over all primes $p\leq x$ with $p \equiv 3 \bmod 4$
Let us define the following functions:
\begin{equation*}
\small A(x)=\prod_{\substack{p\leq x\\ p\equiv 3 \bmod 4}} \Big(1-\frac{1}{p}\Big), \mbox{ } \mbox{ }
B(x)=\prod_{\substack{p\leq x\\ p\e …
- 3,098
3
votes
0
answers
234
views
Asymptotic expansions for the continued fraction $[1,x,x^2,x^3,\cdots]$
The $n$-th convergent is defined as
$$R_n(x) = \frac{P_n(x)}{Q_n(x)}=[1;x,x^2,\cdots,x^n]=1+\frac{1}{x+}\frac{1}{x^2+}\frac{1}{x^3+\cdots}\frac{1}{x^n}$$
where $P_n(x), Q(x)$ are polynomials recursive …
- 3,098
3
votes
2
answers
487
views
Question about a new pseudo-random number generator
While investigating non-periodic RNG's (random number generators) for irrational numbers, I came up with a version that actually produces pseudo-random words consisting of $N$ bits, where $N$ is typic …
- 3,098
3
votes
2
answers
292
views
Does my construction always result in a stationary Poisson point process of intensity $1$? H...
My construction is as follows: Let $X_k$ be a real-valued continuous random variable centered at $k$ (an integer), having distribution $F_k(x,s)$ where $k$ is the location parameter and $s$, a strictl …
- 3,098
3
votes
1
answer
466
views
Curious infinite product, convergence, connection to prime numbers
I have been playing with the following function:
$$
f(x)=\frac{\pi x (1-x^2)}{\sin\pi x}\prod_{k=2}^\infty \frac{\sin(\pi x/k)}{\pi x/k}
$$
It is hard to get correct numerical values. I'll start with …
- 3,098
3
votes
1
answer
572
views
Remarkable limit involving $m_p=\log_p(p^{x_1} + \cdots + p^{x_n})-\log_p(n)$
It is easy to prove that
$\lim_{p\rightarrow 1} m_p = (x_1 + \cdots + x_n)/n$. The following fact about the derivative of $m_p$ with respect to $p$ is also elementary:
$$m'_p =\frac{dm_p}{dp}
=\frac{1 …
- 3,098
3
votes
1
answer
483
views
Simple closed forms for sums such as $\sum_{k=1}^\infty \frac{(-1)^{k+1}}{qk - p}$ and relat...
My goal here is to get a simple expression for $\zeta(3)$. This is a follow up to my previous question posted here. Any Taylor-like expansion from everything I tried won't make it. So this is my last …
- 3,098
3
votes
2
answers
967
views
Recursive random number generator based on irrational numbers
Here $\{\cdot\}$ and $\lfloor \cdot\rfloor$ denote the fractional part and floor functions respectively. For a negative, non-integer number $x$, we use the following definition: $\{x\}=1-\{-x\}$. If $ …
- 3,098
3
votes
0
answers
200
views
Infinite partial fraction expansions to compute fractional iterations and recurrences
Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how …
- 3,098
3
votes
1
answer
685
views
$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?
The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha= …
- 3,098