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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.
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 …
- 188k
2
votes
0
answers
130
views
Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n …
- 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 …
- 82.7k
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
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 …
- 146
2
votes
2
answers
295
views
Convergence of series related to partial fraction expansion of cotangent function
I am looking at the convergence of the series
$$ \cos(t\theta) = \frac{\sin(\pi t)}{\pi} \cdot \Bigg[\frac{1}{t} + 2t \sum_{k=1}^\infty (-1)^k \frac{\cos(k\theta)}{t^2 - k^2}\Bigg].$$
Here $t\in\mathb …
- 6,377
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
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
1
vote
1
answer
416
views
Generalized random harmonic series
Let $Z_n=\sum_{k=1}^n a_k X_k$ with $(a_k)$ a strictly decreasing sequence of positive real numbers that tend to zero. The random variables $X_k$ are independent and satisfy $P(X_k=1) =p_k, P(X_k=-1)= …
- 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
2
votes
1
answer
486
views
Truncated Euler products, Dirichlet eta function, and convergence issues
Can you prove that the following series does not converge if $\frac{1}{2}<\sigma<1$, no matter how close to $1$ sigma is, and no matter how large $t>0$ is? The series is defined as
$$W(\sigma,t)=\sum_ …
- 3,098
12
votes
1
answer
954
views
Strange behavior of $x_{n+1}=x_n +\lambda \sin x_n$
Consider a sequence $(x_n)$ satisfying $x_{n+1}=x_n +\lambda \sin x_n$.
You would expect the sequence $x_n$ to depend on $x_0$ and to exhibit a chaotic, Brownian-type behavior, and indeed it does pret …
- 3,098
1
vote
1
answer
2k
views
About the coefficients of Taylor series for the complex Riemann Zeta function $\zeta(s)$
The following real-valued functions are closely related to the zeros of $\zeta(s)$ in the critical strip $\frac{1}{2}<\Re(s) < 1$.
$$\phi_1(\sigma, t) = \sum_{n=1}^\infty (-1)^{n+1}\frac{\cos(t\log n) …
- 1,126
0
votes
1
answer
277
views
Infinite products for linear combinations of sines or cosines
There is a well known infinite product both for $\phi(x)=\sin x$ and $\phi(x)=\cos x$. These are particular cases of the Weierstrass factorization theorem. What about
$\phi(x)=a_1\cos b_1 x + a_2\cos …
- 91.8k
2
votes
1
answer
258
views
Squaring a semi-convergent series
Let $S=\sum_{n=1}^\infty a_n$, be a semi-convergent series with $T=\sum_{n=1}^\infty a_n^2 < \infty$ and $\sum_{n=1}^\infty |a_n|=\infty$. Under which conditions are the following formulas valid? They …
- 128k