New answers tagged sequences-and-series
0
votes
Formulas for partial composed product
Thanks everyone for the comments! I will try to summarize what's going on a bit, and post it as an answer, as I was not really aware of how symmetric squares and exterior squares work, so I will ...
- 1,091
0
votes
Abel–Plana formula with fractional offset
I eventually did find a published derivation of the fractional-offset (2) of the Abel-Plana formula, in A Generalized Mode Summation Formula of the Zero-Point Energy in a Cavity by Norio Inui (2003):
...
- 172k
0
votes
Abel–Plana formula with fractional offset
https://www.semanticscholar.org/paper/The-Euler-Maclaurin-formula-revisited-Elliott/43a86582da7acc57944cee260d1654bfbc5f251c
A while ago I was looking at using similar formulas for numerical ...
- 270
6
votes
Accepted
Series convergence if $\sum a_n^2 < \infty$
I think the answer is no.
By functional analysis, we know that it suffices to show the existence of a sequence $a_n$ such that $\lVert a \rVert_2 = 1$ and
$$\sum_{n = 1}^{\infty} \left(\sum_{j = 1}^\...
- 354
11
votes
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$
Observe that the left-hand side is the sum of two convergent series. Let $N\geq 2$ be an integer tending to infinity. Truncate the first series at the $N$-th term and the second series at the $2N$-th ...
- 94.4k
5
votes
Natural density of thickly syndetic set
No, the density might not exist, and the upper density (just $\limsup$ in your definition) could be arbitrarily small. Fix any $k$ and define $S = \{a \cdot k^{2b} + c \ : \ a,b \in \mathbb{N}, 0 \leq ...
- 2,233
2
votes
Maximization of $\ell^2$-norm
$\newcommand{\n}{\lfloor{r/c}\rfloor}$We have
\begin{equation*}
s_{r,c}=\sqrt{\n c^2+(r-\n c)^2}.
\end{equation*}
Indeed, by continuity and interchangeability of the coordinates,
\begin{equation*}
...
- 110k
3
votes
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Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$?
For each $n$ we set $q_n = \frac{\text{smallest number more than $nr$ coprime to $n$}}n$. Note that the next prime following $nr$ or the next prime following it is coprime to $n$ for large enough $n$ (...
- 1,555
2
votes
Efficiently computing $\sum_k x^{k^2}$ modulo $p$
Not exactly an answer, but should provide a strong hint on where to look.
This is closely related to Gauss quadratic sums, expressions of the form
$$
g(a;p) = \sum\limits_{k=0}^{p-1} \omega_p^{ak^2},
$...
- 1,091
5
votes
Accepted
Uniqueness of the $J$ invariant
Any meromorphic modular function of weight $0$ for $\mathrm{SL}(2,\Bbb Z)$ is a rational function of $j$, say $P(j)$. Since your function is holomorphic, $P$ is a polynomial. Since your function has a ...
- 35.7k
10
votes
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How can I evaluate the following sum?
The corresponding infinite sum is related to the "incomplete theta function",
$$\Theta_0(x,y)=\sum_{n\geq 0}x^ny^{n(n-1)/2}.$$
We have
$$\sum_{n\geq 0}a^{n^2}=\Theta_0(a,a^2).$$
Accordingly, ...
- 87.1k
7
votes
Accepted
Convergence of derived series
The answer is yes. Indeed, for integers $j\ge0$, let
\begin{equation*}
N_j:=\{2^j,\dots,2^{j+1}-1\},\quad
A_j:=2^{-j}\sum_{k\in N_j}a_k,
\end{equation*}
so that $A_j$ is the arithmetic mean ...
- 110k
2
votes
Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $
You might try to regularize the sum,
$$S(\alpha)=\sum_{n=1}^\infty \frac{ \cos(\alpha n) \sin(n+1) }{n}=
-\tfrac{1}{4} i \left[e^{-i} \ln \left(1-e^{i (\alpha-1)}\right)+e^{-i} \ln \left(1-e^{-i (\...
- 172k
1
vote
Accepted
Implementing the $\pi$ BBP algorithm
Bounding this sum is not hard, it is majorized by a geometric progression.
If you forget about the summands starting from $n + p$ (in your terms this means $s = n + p - 1$)
$$\sum_{k = n + p}^{\infty} ...
- 296
6
votes
Asymptotic for Ramanujan's $\tau$-function
I would like to complement the other answers to mention a couple facts about the erratic nature of the signs of $\tau(n)$. One result is the result of Wilton saying that
$$ \sum_{n \leq N} \frac{\tau(...
- 4,439
4
votes
Slick proof of Stirling's Formula?
I've played around with this a bit. I have a slick lower bound, but not a slick upper bound.
We start with the $\Gamma$-integral:
$$n! = \int_{x=0}^\infty x^n e^{-x} dx = \int_{y=-n}^\infty (n+y)^n e^{...
- 149k
10
votes
Accepted
Asymptotic for Ramanujan's $\tau$-function
While an asymptotic for $|\tau(n)|$ does not exist, there are many results that help us to nail down the order of $|\tau(n)|$. First, let us write
$$\tau(n)=n^{\frac{11}{2}}f(n).$$
Also, let $d(n)$ ...
- 5,014
2
votes
Asymptotic for Ramanujan's $\tau$-function
I am not too familiar with this stuff but it seems highly unlikely Ramanujan's tau function would have general asymptotics due to its arithmetic functional quality. Though you could probably find some ...
- 2,466
9
votes
Asymptotic for Ramanujan's $\tau$-function
No, $\tau(n)$ fluctuates wildly, and it cannot be described in simpler terms. It is "irreducible arithmetic data", and we just love that. Same for its absolute value.
- 94.4k
6
votes
Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?
This isn't a full answer but gives another surprising connection between the two constants. One has
$$\gamma = \int_1^\infty \frac{1-\{x\}}{x^2} dx$$
and
$$\log \frac{4}{\pi} = \int_1^\infty \frac{\...
- 4,064
-1
votes
Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?
This is not an answer per se, but some additional insight. It seems that in certain context there is a meaning in a set of integers with period $2e^{-\gamma}$.
The Chow's EL-numbers are the numbers ...
- 9,294
3
votes
Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed
We may take
$\alpha=\sum_n(1/73)^{f(n)}$
where $f(n)$ is any nonperiodic function.
The prime numbers $2$ and $3$ are both quadratic residues $\bmod 73$, hence products of powers of these primes will ...
- 1,316
5
votes
Accepted
Is anything known about the power series $\sum x^p$ for $p$ prime?
Indeed these functions support a large family of lambert series based non analytic continuations. Around $0$ we have that
$$ \sum_{n=1}^{\infty} x^p = x^2 + \frac{x^3}{1-x^2} - \frac{x^9 +x^{15} + x^{...
- 2,233
9
votes
Accepted
Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed
$\alpha =\sum_{n=1}^{\infty} \frac{1}{ 6^{100^n}}$ should do the trick. A positive proportion of numbers on your list are of the form $2^a 3^b$ for $a,b$ within a reasonable constant factor of each ...
- 131k
4
votes
A vanishing sum and related $p$-adic congruences
The Archimedean version $(1)$ can be somehow systematically proved using WZ-method.
We recall the following analytic fact (c.f. Propositon 2.1): if $F,G$ satisfies $F(n+1,k) - F(n,k) = G(n,k+1)-G(n,k)$...
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