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1
vote
0answers
207 views

Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ contains arbitrary long arithmetic progressions?

This is somewhat related to Erdős conjecture on arithmetic progressions Is there an infinite sequence of positive integers $a_n$ s.t. $\sum_{n=1}^\infty{\frac{1}{a_n}}$ converges and $a_n$ ...
6
votes
2answers
412 views

Alternating sums of GCDs

The exciting question on alternating sums of binomial coefficients triggered me to ask the following much simpler question (sorry if it is too simple, but I'm not a number theorist, so I must be ...
36
votes
2answers
3k views

Alternating sum of square roots of binomial coefficients

Let $$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
63
votes
7answers
5k views

Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$? (I think it is.) If so, how would one prove this? (To confirm: This is the power series for $e^x$, except with the ...
3
votes
1answer
307 views

Products of trigonometric functions with increasing frequencies

I am looking at weighted $L_2$ norms of a class of Littlewood polynomials, related to Walsh and Rademacher functions which made me look for pseudo-closed forms or computationally efficient expressions ...
4
votes
1answer
417 views

Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

I am not familiar with newforms, so this may not make any sense. OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3] Numerical evidence suggest that up to $10^5$ $$ ...
14
votes
0answers
802 views

Regularizing the divergent sum $1^k + 2^k + \cdots$

EDIT: Under this "regularization", the harmonic series can be interpreted as $s_{-1}$ and assigned a value $$s_{-1} = \lim_{k \rightarrow 1} \zeta(k) (2-2^k) = -2 \log 2$$ I was looking at ...
5
votes
2answers
751 views

Divergence of Dirichlet series

Suppose $s$ is a complex number with $\Re(s) \in (0,1]$ and $\{a_n\}$ is a complex sequence converging to $a \neq 0$. Must the Dirichlet series $$\sum_{n=1}^\infty\frac{a_n}{n^s}$$ diverge? I asked ...
4
votes
0answers
741 views

Has n^2*|sin(n)| limit? [closed]

Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity. In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...
9
votes
2answers
1k views

Defining the slowest divergent series

This question might seem too fuzzy, and if so, I will be happy to withdraw it. Until then, here it is: I know that a method of slowing a divergent series of positive reals is to replace the $n$-th ...
2
votes
2answers
877 views

Ratio of Sequences Sum Inequality

I have two real sequences $a_1,a_2,\dots,a_n$ and $b_1, b_2, \dots, b_n$, with $a_i > 0$ and $1 \leq b_i < n$, and I'm looking for a lower bound of $\sum_i \frac{a_i}{b_i}$ in terms of $\sum_i ...
1
vote
2answers
321 views

Asymptotics of Fourier coefficients of power-type functions

I would like to understand the asymptotic behaviour of the Fourier coefficients of power type functions $f(t) = |t|^{-\alpha} 1_{[-\pi, \pi]} \qquad 0 < \alpha<1.$ I suppose this is a classic ...
1
vote
1answer
214 views

exponential sum - general cases [closed]

hello community. I need some help given $k = \sum A_q e^{ia_q s}$ where $k$, and $s$ is known. Can $A_q$ s now be expressed in terms of $a_q$ . any help in that direction will be appreciated.
4
votes
0answers
151 views

Are there infinite sequences of rational cubes whose first differences are positive squares?

This is related to How many sequences of rational squares are there, all of whose differences are also rational squares? Are there infinite sequences $a_n$ of rational cubes whose first ...
1
vote
1answer
306 views

Limit of an infinite series, related to (generalised Newton's) binomial expansion

How to calculate the infinite sum of the following series, related to binomial expansion for rational number, $r$: ...
2
votes
0answers
376 views

How to calculate/approximate expectation of function of a binomial random variable?

Hi, I am stuck at following problem in my research. Suppose that $M=m$ is a random variable with binomial distribution with parameters $n,p$. The constants $r$ and $\gamma$ are greater than zero. ...
0
votes
2answers
243 views

Summing a function using modulus. [closed]

The problem: If the infinite sum of a function is known, how to find: $$\begin{align*} \sum_{i\equiv 0 \mod m}f(x_0+i)=\\\\ f(x_0)+f(x_0+m)+f(x_0+2m)+f(x_0+3m)+\ldots \end{align*}$$ And if the ...
1
vote
0answers
262 views

Sequences defined as solutions to equations : $u_{n}=v_{n}^n$ [closed]

Hello, I hope this is the appropriate forum for posting. For $n$ a positive integer at least equal to $2$, define the two following functions as follows : $f_{n}(x)=\pi/4 + ...
2
votes
2answers
330 views

Inverting an asymptotic series

I have the first few terms of a series of the form, $y(x)=\ln(x)+x+a_0+\frac{a_1}{x}+\frac{a_2}{x^2}+\cdots$. Knowing that the inverse $x(y)$ exists, I am looking for method to write x in terms of ...
2
votes
2answers
830 views

Sum of products of exponentials and polynomials

Hi, I am looking for a closed-form expression for the finite sum of the product of an exponential function with a polynomial function --- that is, the sum ...
1
vote
0answers
212 views

Applying the ideas of power series to certain convolutions - which identities transfer?

Let's suppose I'm working with some set of functions $f_k(n)$. $f_1(n)$ is essentially the root of my functions, and could be nearly anything, and then $f_k(n) = (f_1(n) * f_{k-1}(n))$ for some ...
2
votes
1answer
308 views

differential operator power coefficients

Let $(F(x)\frac{d}{dx})^n=\sum_{i=1}^n H_{n,i}(F, F', F^{(2)}, \ldots , F^{(n)})\frac{d^i}{dx^i}$. I'm curious about the exact formula for $H_{n,i}(y_0, y_1, \ldots , y_n)$. What is known about it?
4
votes
2answers
603 views

Proving a sequence of integrals increases (iterated minimax distributions)

I am trying to show that $$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = (1-(1-F_{n-1}(x))^c)^c$$ and $F_0(x) = x$ and $n$ and $c$ are integers, $n\geq 1$ and $c \geq 2$ Note that ...
5
votes
0answers
721 views

Convergent series of primes [closed]

If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
2
votes
2answers
606 views

Sum of Series Where Exponent is Sum of Arithmetic Progression

Hi, How do i get the sum of such a sequence: $1 + x^{-1} + x^{-3} + x^{-6} + ...$ where the exponents are actually sum of arithmetic progression. i.e. $x^{-0} + x^{-(0 + 1)} + x^{-(0 + 1 + 2)} + ...
-3
votes
2answers
646 views

An interesting, simple, sequence - surprised to find little material. [closed]

I've been considering this sequence: $$1,2,3,6,12,24,48,96,192,...$$ I've generated the sequence from the rule $$V_n=\sum_{0\leq i \lt n} V_i$$ $$V_0=1; V_1=2V_0=V_0+V_0$$ What interests me most, ...
6
votes
2answers
245 views

Repeatedly indexing into an $\infty$-sequence of integers

Suppose one has in hand an infinite sequence $s$ of distinct natural numbers, for example, $$s=s_1=(1, 3, 5, 7, 9, 11, 13, 15, 17, 19,\ldots) \;.$$ So this sequence can be considered an injection ...
2
votes
0answers
500 views

What square-summable sequences are “sinc-summable”?

$\operatorname{sinc} : \mathbb{R} \to \mathbb{R} \;\;$ is defined by $\;\; \operatorname{sinc}(x) \; = \; \begin{cases} 1 & \text{if }\:\;x=0 \\ \\ \frac{\operatorname{sin}(x)}x & \text{else} ...
6
votes
2answers
750 views

Is there a combinatorial interpretation of this triangle sequence? Is there a “simpler” formula?

Hi. I was wondering about this. I've got this triangle sequence: $$a_{n,k} = \sum_{1 = m_1 < m_2 < \cdots < m_k = n}\ \prod_{j=2}^k S(m_j, m_{j-1})$$. where the $S(n, k)$ are Stirling ...
2
votes
1answer
244 views

The relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$

During my research I have recently stumbled upon the problem of finding the relative homotopy sets $\pi_1(U_n,U_n/O_n)$ and $\pi_1(U_{2n},U_{2n}/USp_{2n})$ for $n$ large enough to be in the stable ...
20
votes
3answers
2k views

The Wronskian of sin(kx) and cos(kx), k=1…n

What is the determinant of the Wronskian of the functions $\{\cos\ x, \sin\ x, \cos\ 2x, \sin\ 2x,\ldots, \cos\ nx, \sin\ nx\}$? This determinant seems to be an integer, and the sequence starts with ...
8
votes
2answers
809 views

What is the series expression for (1+1/x)^x about x = \infty?

This seems like it must have been addressed somewhere already, but I cannot find it in any standard series tables. I have the equation: $f(z) = \left(1 + \frac{1}{z}\right)^z$. What is the general ...
4
votes
2answers
305 views

Is there a sufficient criteria to guarantee that $\lim_{n} a_{nn} = \lim_{m} \lim_{n} a_{mn}$ ?

Let $a_{mn}$ be a sequence in some $\mathbb{R}^k$. We know in advance that $$\lim_{n} ~a_{nn} = L_1, \qquad \lim_{m}~ \lim_{n} ~a_{mn} = L_2 $$ exist. Is there a sufficient criteria to conclude ...
2
votes
3answers
2k views

Elementary proof of the equidistribution theorem

I'm looking for references to (as many as possible) elementary proofs of the Weyl's equidistribution theorem, i.e., the statement that the sequence $\alpha, 2\alpha, 3\alpha, \ldots \mod 1$ is ...
2
votes
2answers
187 views

Distribution of $\log_\mu n$ mod 1

If $\langle x\rangle $ is the fractional part of $x$, it is known that for $0<\mu<1$, the sequence $\langle \log_\mu n\rangle _{n=1}^\infty$ is dense in $[0,1]$ but is not uniformly distributed. ...
7
votes
2answers
764 views

Exact Value of a Series

It is very easy to show that the series $$\frac{1-1/2}{1\times2} - \frac{1-1/2+1/3}{2\times3} + \frac{1-1/2+1/3-1/4}{3\times4} - ...$$ i.e. $$\sum_{n=1}^{\infty} ...
4
votes
0answers
197 views

Number of times lead changes in a multi-candidate election (reference-request)

In a two candidate election where votes are distributed uniformly at random between the candidates, the probability that the lead changes when tallying the $i$-th vote is the same as the probability ...
8
votes
4answers
2k views

Unique limits of sequences plus what implies Hausdorff?

It is known that there are non-Hausdorff spaces which admit unique limits for all convergent sequence (see here) and it is also known that unique limits for nets implies Hausdorff. What I am ...
18
votes
3answers
1k views

Zeroes of the random Fibonacci sequence

Let X_n be the "random Fibonacci sequence," defined as follows: $X_0 = 0, X_1 = 1$; $X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips. It is known that ...
0
votes
1answer
186 views

Distribution wanted

I have a centered random variables $Y_1,Y_2$ which first 10 moments are given respectively by $$ 0, 1, 0, 6, 0, 90, 0, 2520, 0, 113400 $$ $$0, 1, 0, 32/3, 0, 36847/100, 0, 436879364/15435, 0, ...
0
votes
0answers
148 views

Asymptotic behaviour of a recursively defined sequence

I encounter a problem in which I need to characterize the asymptotic behaviour of a sequence. $\{s_{n,k}\}$ is a stationary probability distribution, i.e., ...
2
votes
4answers
998 views

Closed-form for modified formal power series

This question have been driving me crazy for months now. This comes from work on multiple integrals and convolutions but is phrased in terms of formal power series. We start with a formal power ...
3
votes
0answers
678 views

Method for variable substitution in multiple summation

I want to ask: is there any general method for variable substitution in multiple summation? For example in the following equation a new variable $\lambda=n+m-2\mu$ is introduced to transform the LHS ...
0
votes
0answers
277 views

Infinite Product

Hi there, It has been long time since I had my math classes. However, I am trying to find a close form expression (if exists) of the following infinite product $f(n) =\prod\limits_{i=n}^{\infty} ...
4
votes
2answers
609 views

Asymptotics for the number of ways to sum primes such that the sum is <= n

Hello! Given $n$ I would like to find a lower bound (or a tight asymptotics) for the number $s(n)$ of solutions to $$ p_1 + \ldots + p_k \leq n \quad (1) $$ where $k$ is arbitrary and $p_1 \leq ...
2
votes
0answers
466 views

recursive sum of products of bessel functions

I have found a way to sum products of bessel functions in the form $$S_\ell(x,y)=\sum_{n=-\infty}^\infty (-1)^{n+\ell} I_{\ell-2n}(x)I_n(y)$$ recursively, i.e. once $S_0(x,y)$ is found, via the ...
0
votes
0answers
207 views

Square variation norm and non-negative, non-increasing sequences

I am trying to understand the properties of square variation, namely, the possibility of preserving it under certain operations. I am following Albiac & Kalton's book: Let $J$ stand for the usual ...
2
votes
1answer
314 views

Can averaged limits of sequences be realized as limits of sequences?

Let a summation method take a sequence $(x_n)$ to a net $(y_\alpha)$, where $\alpha$ runs over a partially ordered set, $y_\alpha=\sum c_{\alpha,n}x_n$ ($c_{\alpha,n}\geq 0$, $\sum_n c_{\alpha,n}=1$ ...
17
votes
2answers
2k views

Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?

Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and $$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$ See ...
9
votes
3answers
2k views

Ramanujan's eccentric Integral formula

The wikipedia page on Srinivasa Ramanujan gives a very strange formula: Ramanujan: If $0 < a < b + \frac{1}{2}$ then, $$\int\limits_{0}^{\infty} \frac{ 1 + x^{2}/(b+1)^{2}}{ 1 + ...