All Questions
34 questions
122
votes
5
answers
27k
views
Is the series $\sum_n|\sin n|^n/n$ convergent?
Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent?
(The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
- 4,535
23
votes
4
answers
2k
views
Identity for an infinite product
Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes".
QUESTION. Is this true?
$$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
- 43.2k
18
votes
5
answers
3k
views
Bernoulli sum meets golden number
Let $B_n$ denote the Bernoulli numbers and let $\phi=\frac{1+\sqrt{5}}2$ be the golden ratio.
I encountered the following infinite sum and would like to ask:
Question. Is this true? If so, any ...
- 43.2k
15
votes
4
answers
3k
views
No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
13
votes
3
answers
1k
views
iterated harmonic numbers vs Riemann zeta
Define the $m$-th iterated harmonic sums in the manner: $\bar{H}_0(n):=1$ and for
$m\geq1$ by
$$\bar{H}_m(n):=\sum_{k=1}^n\frac{\bar{H}_{m-1}(k)}k.$$
For example, $\bar{H}_1(n)=\sum_{k=1}^n\frac1k$ ...
- 43.2k
13
votes
3
answers
810
views
Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?
For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$
the sum of remainders function, the arithmetic function A004125 from the OEIS.
Example. We'...
11
votes
2
answers
587
views
Extracting a subsequence common to infinitely many sets from an uncountable collection with uniform positive upper density
Let $\{a_n\},\{b_n\}$ be strictly increasing sequence of positive integers satisfying $a_1<b_1<a_2<b_2<a_3<b_3<\ldots$ and $(b_n-a_n) \to \infty$. Define $I_n:= [a_n,b_n]$, meaning ...
- 271
11
votes
1
answer
430
views
Cantor set intersecting a geometric sequence
I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
- 113
9
votes
2
answers
440
views
How to prove this sum involving powers of cosec is an integer?
It is claimed that the following function produces only integer values for all integer $m \geq 1$, $N \geq 2$.
$F(m,N)=\frac{N^m}{2^m}\displaystyle \sum_{j=1}^{N-1} \operatorname{cosec} ^{2m}\left(\...
- 201
7
votes
1
answer
507
views
Is the mapping $f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}$ surjective?
Is the mapping
$$
f: \mathbb{R} \rightarrow [0,1], \ x \mapsto \sum_{n=1}^\infty \frac{\lfloor x^n \rfloor \mod 2}{2^n}
$$
surjective?
If not, what is its image?
If yes, what can be said about ...
- 19.6k
7
votes
1
answer
1k
views
Signed variant of the Flint Hills series
I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one:
$$
\sum_{n = 1}^\infty \...
- 17.2k
6
votes
1
answer
392
views
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]
I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
6
votes
3
answers
536
views
A need for analytic continuation of a finite sum function
Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$.
I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum)
\begin{align*}
{\color{red}...
- 43.2k
6
votes
1
answer
223
views
Asympotic density of a very simple sequence
Let $A=\{mn(m+n)\mid n,m\in \mathbb{N}_0\}$. Sorted, this is OEIS sequence A088915. What is its asymptotic behavior? It seems approximately $a(n)=O(n^{1.5})$, but not quite.
I'm actually even more ...
- 5,103
6
votes
0
answers
283
views
Is the arithmetic-geometric mean of 1 and 2 rational?
It is easy to show that, for two fixed real numbers $\alpha, \beta > 0$, the sequences given by $a_ 1 = \frac{\alpha + \beta }{2}$, $ g_1 = \sqrt{\alpha\beta}$, and $a_{n+1} = \frac{a_n + g_n}{2}$, ...
- 161
6
votes
0
answers
267
views
Convergence of $\sum_{n=1}^\infty x_n^k$
I thought that this question is more suitable for MSE, and asked it there. (Link to the MSE question) However, it does not get any answer despite the upvotes. It appears that I might have ...
- 1,755
5
votes
2
answers
874
views
Searching for a proof for a series identity
The below identity I have found experimentally.
Question. Is this true? If so, may you provide a "slick" (or any) proof.
$$6\sum_{k=1}^{\infty}\frac{k^2q^k}{(1-q^k)^2}+12\left(\sum_{k=1}^{\infty}...
- 43.2k
5
votes
1
answer
680
views
When does this interesting sum diverge?
For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$
I don't know of any references or methods for this -- not even for $x=1$, for which the ...
- 3,443
5
votes
0
answers
285
views
How do you go about making ranges (for integer variables) independent?
Basic question: say you have a sum
$$\sum_{n_1 n_2 \dotsb n_k \leq x} f(n_1,\dotsc,n_k),$$
where $f$ decomposes in some sense (say: $f(n_1,\dotsc,n_k) = g(n_1) + \dotsb + g(n_k)$, or $f(n_1,\dotsc,n_k)...
- 20.2k
5
votes
0
answers
343
views
Can the inverse of the Riemann zeta function in $s > 1$ be expressed as a series?
In this post, we are interested in the Rimenann zeta function $\zeta(s)$ in $s > 1$ only where it is strictly decreasing rather than $s$ in the entire complex plane. We have the Stieltjes series ...
- 5,470
4
votes
1
answer
254
views
$\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}$ for various $x$
Let $$f(x)=\limsup_{n\rightarrow \infty, n\in\mathbb{N}} \sin(n)^{n^x}.$$
Compute $f(1)$ and $f(2)$.
4
votes
1
answer
188
views
Is there a generalization of these q-series identities?
Denote $(q;q)_n=(1-q)(1-q^2)\cdots(1-q^n)$.
The below three identities are known.
\begin{align*}
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\binom{n+1}2}}{(q;q)_n}
&=1-\sum_{n\in\mathbb{Z}}(-1)^nq^{\...
- 43.2k
3
votes
1
answer
702
views
$\{(\log n)^\alpha\}$ not equidistributed if $0<\alpha\leq 1$, so how is it distributed?
The brackets denote the fractional part function. It is well known that the distribution (defined as the limit of the empirical distribution) is $F(x)=(e^x - 1)/(e-1)$, with $x\in [0, 1]$, if $\alpha=...
- 3,098
3
votes
1
answer
631
views
Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?
This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even ...
- 128k
3
votes
1
answer
367
views
Convergence of a triple sum involving the imaginary part of the Riemann zeta function's non trivial zeros
Let $N>0$ an integer, $k>0$ a real parameter and let $\rho = \beta +i \gamma$ a non trivial zero of the Riemann zeta function. For a work I need to find the best possible $k$ such that $$I=\sum_{...
- 219
2
votes
2
answers
261
views
Prove a family of series having integer coefficients
I encountered a certain family of infinite series in some work, which is given by
$$F_r(x)=\frac1{2^r}\sum_{k=0}^r\binom{r}k\frac1{1+x(2k-r)^2}.$$
I've convincing date to believe the following is true,...
- 43.2k
2
votes
1
answer
332
views
Convergence of $\sum(n^p\sin^qn)^{-1}$
I've been recently interested in the problem of convergence of the function in such form: $\displaystyle \sum_{n=1}^\infty\frac1{n^p\sin^q n}$.
I saw there's been discussion here when $p=3, q=2$ and $...
- 21
2
votes
1
answer
192
views
Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$? [closed]
This seems pretty trivial but I can't seem to figure it out. I think it's obviously true, given an unconstrained convergent sequence we just have to add some filler elements, but I'm having trouble ...
- 291
2
votes
0
answers
120
views
A sequence linked to irrationality
Let $0 < c < 1$ be a real number and $ x \in \mathbb{R}$. We define the sequence $(u_n)_{n \in \mathbb{N}}$ by :
$$u_0 = x$$
$$ \mathrm{If}, u_n \le c, \mathrm{then}, u_{n + 1} = u_n + (1 - c) $$...
- 69
2
votes
0
answers
448
views
Conjecture: The sequence {$π(2n+1)!$} is equidistributed in the interval (0,1)
Let $n\in\mathbb{N}$.
From the book "Uniform Distribution of Sequences" (available here) by L. Kuipers and H. Niederreiter, (from pg. 8) I found that for any irrational $\theta$, the ...
1
vote
1
answer
142
views
About a Dirichlet series [closed]
I would like to know if the following assertion is true:
Let consider a real decreasing sequence $(t_n)$ of positive numbers with limit zero, if the series $\sum\limits_{n=1}^\infty(t_n)^a$ is ...
- 337
1
vote
0
answers
175
views
Solution of recurrence relation with summation
I have the following recurrence relation:
$$b(n,k)=\sum _{\text{i}=0}^{2 n-1} \left(b(n-1,k-\text{i})+\frac{\text{i} (2 n-\text{i}) \binom{2 n-1}{\text{i}} \binom{(n-2)^2}{k-\text{i}}}{2 n-1} \right)$$...
- 47
1
vote
0
answers
99
views
simultaneous smallness
QUESTION. Given reals $0 < \epsilon, \delta < 1$, is it always possible to find $m, n \in \mathbb{N}$ such that
$$\begin{cases} \qquad \,\,\,\, \,(1-\delta^m)^n < \epsilon \\
1-(1-(\frac{\...
- 43.2k
-1
votes
1
answer
173
views
For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$
Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...
- 11