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24
votes
2answers
897 views

Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
4
votes
1answer
129 views

On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer? Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence ...
1
vote
1answer
167 views

Expression for infinite sum of two Bessel functions and a power

I'm searching for an expression of the following sum (all indices are integers and all variables are positive real numbers): $$\sum_{\lambda=-\infty}^{+\infty} A^{|l-\lambda|} B^{|\lambda|} ...
3
votes
0answers
97 views
+50

Asymptotic behavior of a sequence of functions

For $n\in\mathbb{N}$ and $q\in(0,1)$, define $$f_{n}(q):=\sum_{i_{1},i_{2},\dots,i_{n}=1}^{\infty}\frac{q^{i_1+i_2+\dots+i_n}}{(1-q^{i_1+i_2})(1-q^{i_2+i_3})\dots(1-q^{i_{n-1}+i_n})(1-q^{i_n+i_1})}.$$ ...
-2
votes
0answers
52 views

The exponents, sequences-and-series [closed]

If we have the formula $\frac{n^m }{m}$ ~ $({1 ^ {m-1}} + {2 ^ {m-1}}+...+ {(nb) ^ {m-1}}) * a^{m-1}$, if $a$ = $\frac{1 }{10^x}$ ( $x$ can be any number $1,2,3..$) and $b$ = $\frac{1 }{a}$, ...
21
votes
4answers
1k views

Does this sequence always give an integer?

It is known that the $k$-Somos sequences always give integers for $2\le k\le 7$. For example, the $6$-Somos sequence is defined as the following : $$a_{n+6}=\frac{a_{n+5}\cdot a_{n+1}+a_{n+4}\cdot ...
1
vote
1answer
63 views

Local Uniform Convergence

Suppose $f(x)$ is a positive continuous function on $[0,\infty)$ and that $f(x+u)-f(x)\to 0$ as $x\to\infty$ for every given $u\in[0,\infty)$. Prove that, given any $a>0$, $f(x+u)-f(x)\to 0$, as ...
0
votes
0answers
29 views

Fractal in discrete time series/discrete time sequence

Consider a time series of real number $x_1, x_2,\dots,...x_n$. How one can define fractal dimension of this series? I would like to know famous formula $F+H=2$ where H is Hurst exponent and F is ...
2
votes
0answers
69 views

Oscillation aspects of two-way infinite alternating series (a followup from the MO-question “functions that eventually oscillate”)

In the recent question on "eventually oscillating function" I had a heuristic for the function $d(x)$ that its amplitude is constant, but could not further describe that function. I just found a ...
30
votes
1answer
581 views

Rearrangements of a power series at the boundary of convergence

Take some power series $f(z) = \sum a_n z^n$ with a finite non-zero radius of convergence. I can rearrange the terms of the series, say, to get a different infinite series $$f_{\sigma}(z) = ...
2
votes
1answer
413 views

How to find the coefficients of a poor-converging series?

I have the series $\psi(r,\theta;p)=\sum_{n=0}^{\infty} a_n J_{\pi n/\Phi}(p r)\cos(\pi n \theta/\Phi)$ and the boundary conditions $\psi(r,\pm\Phi;p)=\sum_{n=0}^{\infty} a_n J_{\pi ...
4
votes
1answer
365 views

Ramanujan's series for $(1/\pi)$ and modular equation of degree $29$

In his famous paper "Modular Equations and Approximations to $\pi$", Ramanujan gives the following famous series for $1/\pi$: \begin{align}\frac{1}{2\pi\sqrt{2}} &= \frac{1103}{99^{2}} + ...
9
votes
1answer
448 views

Does every strictly increasing, unbounded sequence of positive real numbers contain arbitrarily long, finite subsequences which are “sort of increasing” or “sort of decreasing” (as defined below)?

Is the following true? If $(x_0, x_1, \dots)$ is a strictly increasing, unbounded sequence of positive real numbers, then there exist fixed $M,N \geq 1$ such that the sequence $(x_0, x_1, \dots)$ ...
5
votes
3answers
249 views

Sets of natural numbers with finite intersections and divergent sums of reciprocals

Does there exist an uncountable collection $\Lambda$ of infinite subsets of the set of natural numbers such that (i) any two distinct subsets in the collection have a finite intersection and (ii) the ...
5
votes
0answers
113 views

How to assess the influence of a specific term in this telescoping series for $\zeta(s)$?

I like to expand on this (unanswered) MSE question. Take the following, nicely symmetrical, telescoping series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{2\,(s-1)} \left(1+\sum ...
0
votes
0answers
79 views

An infinite sum which approaches a geometric series

This is a cross-post from http://math.stackexchange.com/questions/1174998/estimating-an-unusual-infinite-sum, which didn't get any useful answers I encountered the following unusual type of infinite ...
3
votes
0answers
101 views

Euler series with milder divergence

Theorema 19 in Euler's memoir "Variae observationes circa series infinitas" says The sum of the reciprocals of the prime numbers is infinitely great but is infinitely times less than the sum of the ...
10
votes
1answer
618 views

Divergence of a series similar to $\sum\frac{1}{p}$

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$. We denote the least integer not covered by the arithmetic ...
11
votes
2answers
457 views

What is the Hausdorff dimension of this fractal?

Let $\sum_{i=h}^\infty d_i/b^i $ be the base $b$ representation of $x \geq 0,$ where $b>1$ and the $d_i$ are uniquely determined by the greedy algorithm. For fixed $c>1,$ let $f(x)= ...
0
votes
3answers
140 views

Hypergeometric sum specific value

How to show? $${}_2F_1(1,1;\frac{1}{2}, \frac{1}{2}) = 2 + \frac{\pi}{2} $$ It numerically is very close, came up when evaluating: $$ \frac{1}{1} + \frac{1 \times 2}{1 \times 3} + \frac{1 \times 2 ...
5
votes
1answer
186 views

Asymptotic expansion of $\zeta(s \mid a,b)= \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}$

I'm interested in an asymptotic expansion of the following Riemann zeta-type function $$ \begin{align} \displaystyle \zeta(s \mid a,b) := \sum_{n=1}^{\infty} \frac{1}{(n+a)^{s}(n+b)}, \quad \Re a ...
3
votes
1answer
219 views

New series for $1/\pi$ based on Ramanujan's ideas

In his classic paper "Modular Equations and Approximations to $\pi$ (1914)", Ramanujan gives a standard technique to obtain a general family of series for $1/\pi$ based on series for $(2K/\pi)^{2}$ in ...
2
votes
1answer
70 views

Does the green kernel converge as a series of functions?

Let $(M,g)$ be a compact rimannian manifold. It is well known that we can diagonalyse the Green kernel as a $L^2$ operator acting on functions. Moreover we have the convergence of the following ...
3
votes
0answers
216 views

Solving a doubly exponential generating function

I am analyzing the average time complexity of some algorithm on some probabilistic model, and I've come to a doubly exponential sequence for which I cannot find corresponding generating function. I ...
1
vote
1answer
76 views

Series estimate

Let $\theta\in(0,1)$ be given. I define for $a>0$ and $\lambda \ge 1$, $ S(\lambda,a )=\sum_{k\ge 1} k^{\frac12-\theta}e^{-a\vert k-\lambda\vert}. $ I want to prove that $$ ...
10
votes
2answers
227 views

Vanishing of certain periodic series: A question related to $L(1 , \chi) \neq 0$.

Fix $q$ to be a positive integer. Let $$f : \mathbb{N} \to \{-1 ,0, 1\}$$ be a $q$-periodic arithmetic function such that $$\sum_{n = 1}^q f(n) = 0.$$ If $f$ is not identically zero, is it true that ...
35
votes
2answers
1k views

Why does this sequence converges to $\pi$?

One of my daughters was having a small programming exercise. Let's consider following algorithm: Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$. Remove every $2$nd number. From the ...
9
votes
1answer
696 views

How to rearrange only negative part of a conditionally convergent series to get any sum greater then initial?

Suppose that $\sum^\infty_{n=1} u_n = s,$ where the series converges conditionally, and $s'>s$. How to prove the existence of such a permutation $\sigma,$ such that 1) $u_n\geq 0 \rightarrow ...
7
votes
0answers
281 views

About the first decimal of $\sqrt {n!}$

Do we have : $$\sup\{\sqrt {n!} - E(\sqrt {n!}); n\in I\!\!N\}=1?$$ Where $E(\cdot)$ is the integer part function, and $n!=1\times 2...\times n$.
0
votes
3answers
778 views

Simplifying finite sum over 1/(ax+b)

Can I simplify: \begin{equation} \sum_{x=x_0}^{x_1} \frac{1}{ax+b} \end{equation}
7
votes
2answers
451 views

Newton series and Fourier transform - is there an analogy?

Fourier expansion for a function: $$f(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}\int_{-\infty}^{+\infty}e^{i\omega t}f(t)dt \, d\omega$$ Newton series expansion of a function: ...
1
vote
0answers
47 views

closed form for a series with binomials and primes

does the series $\sum_{n=0}^\infty p^n \binom{x}{p^n}$ have a closed form ? ($p$ prime) this is a special case of $\sum_{n=0}^\infty p^n \left(\sum_{k=p^n}^{p^{n+1}-1}a_k\binom{x}{k}\right)$ with the ...
22
votes
7answers
2k views

What problem would you base your mathcoin on?

Recently, a variant of electronic currency, based on prime sextuplets, broke the record in generating the largest known set of six primes, packed as closely as possible, that is, a sextuple ...
28
votes
3answers
1k views

Why do Pell equations appear in Ramanujan's pi formulas?

While answering this MSE question about the Pell equation $x^2-29y^2=1$, I noticed that certain fundamental solutions appeared in Ramanujan's famous pi formula. I. Given the fundamental unit, ...
5
votes
0answers
117 views

Are these identities Newton series?

Newton series is the following expansion of a function: $$f(x)=\sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)=\sum_{n=0}^{\infty} {x\choose n} \sum_{k=0}^n{n\choose k}(-1)^{k-n}f(k)$$ Now ...
4
votes
0answers
188 views

Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas: 1) The ...
1
vote
1answer
181 views

Abscissa of absolute convergence of the product of two Dirichlet series

I first asked the following question on Mathematics StackExchange (a few weeks ago), since the content of MathOverflow is mostly above my academic level. I didn't want to bother people on this forum ...
2
votes
11answers
7k views

Uniquely generate all permutations of three digits that sum to a particular value?

I'm looking for a way of generating all permutations of three digits (actually xyz) that sum to the same value. For example: ...
2
votes
5answers
348 views

Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$ I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$. If I do it iteratively step ...
1
vote
2answers
378 views

How this expression leads to the given sequence

Here given is a sequence from OEIS. The sequence is triangle of coefficients from fractional iteration of $e^x - 1$. Few terms are: 1, 1, 3, 1, 13, 18, 1, 50, 205, 180, 1, 201, 1865, 4245, 2700, 1, ...
0
votes
0answers
107 views

this sequence $A_{n}$ have recursive relations?

Let $$A_{n}=\sum_{i=0}^{n-3}(-1)^{n+i-2}\dfrac{13n^2-31n-10ni+9i+i^2+16}{(3n-i-3)(3n-i-4)(2n-i-3)!\cdot i!}$$ I want find the $A_{n}$ recursive relations,such as following form ...
5
votes
2answers
243 views

expression for infinite series with powers of factorial in denominator

The series $$\sum_{k=0}^\infty \frac{\exp(c k \beta)}{(k!)^\beta} $$ has come up when I'm trying to apply the methodology in this paper (http://www.ism.ac.jp/~eguchi/pdf/Robustify_MLE.pdf) to Poisson ...
12
votes
4answers
666 views

How to calculate the infinite sum of this double series?

I'm calculating this double sum: $$ \sum _{m=1}^{\infty } \sum _{k=0}^{\infty } \frac{(-1)^m}{(2 k+1)^2+m^2} $$ I know the answer is $$ \frac{ \pi \log (2)}{16}-\frac{\pi ^2}{16} $$ which can be ...
17
votes
1answer
857 views

Ramanujan's pi formulas with a twist

Given the binomial function $\binom{n}{k}$. 1. Define the following sequences, $$\begin{aligned} u_1(k) &= \tbinom{2k}{k}\tbinom{3k}{k}\tbinom{6k}{3k} = 1, 120, 83160, 81681600,\dots \\ u_2(k) ...
2
votes
2answers
231 views

Is this infinite series related to some well-known special functions?

Please allow me to resort once again to the expertise of the MathOverflow community : During research I encoutered the following infinite series : $$\sum_{n=-\infty}^{+\infty} ...
30
votes
3answers
3k views

Is there an “elegant” non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?

Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway... Consider this problem. I've been trying to find a formula to expand the "regular iteration" of ...
3
votes
1answer
268 views

Does this function have any exponential growth?

Has anyone seen any function of the following type? $$ g(x):=\sum_{n=0}^\infty \frac{x^n}{n!}\exp\left(-\frac{a^n}{x}\right),\quad a>1,x\ge 0. $$ The question is whether for some constant ...
0
votes
1answer
113 views

Energy of repeated filter

For given sequences $a=(a_1, a_2, \cdots)$ and $b=(b_1, b_2, \cdots)$, define $$a \star b$$ as the convolution. Formally, $$c=a \star b$$ implies the $i$th element of $c$, $c_i$, satisfies the ...
6
votes
3answers
399 views

Asymptotic formulas for Monster-related modular functions?

Define the following, $$j(\tau) = \Big(\tfrac{E_4(\tau)}{\eta^8(\tau)}\Big)^3 = {1 \over q} + 744 + \color{blue}{196884} q + 21493760 q^2 + 864299970 q^3 + \cdots \tag{1}$$ $$j_{2A}(\tau) ...
2
votes
0answers
197 views

Minimizing $\{0,1\}$-sequence permutations

Explanation: For a given bit sequence $f$, reposition the bits as to minimize $G$ which can be thought of as a measure of how poorly proportional $f$ is to each of its subsequences. Let $p \in ...