The sequences-and-series tag has no wiki summary.

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### 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 ...

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247 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 ...

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88 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
...

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64 views

### Handwaving explanation of “unit root” sought [closed]

I am struggling to get my head around the concept of "unit root" in relation to time series. And it would be a great help if someone could give me a two or three sentence handwaving explanation of the ...

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457 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 ...

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**3**answers

734 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,
...

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### 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 ...

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255 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 ...

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**1**answer

234 views

### The closed form of $\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$

The following series I'm interested in $$\sum_{n=2}^\infty(-1)^{n+1}\frac{\psi(n)}n\log(n)$$
where $\psi(n)$ is digamma function
arose in the evaluation of an integral I posted on MSE, ...

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186 views

### A.e. pointwise convergence of L2 functions - counterexample for generalization of Carleson's thm

Let $f_n \in L^2[0,1]$ be an orthonormal sequence and let $c_n \in \mathbb C$ be such that $\sum_{n = 1}^{\infty} |c_n|^2 < \infty$. Does this imply that the sequence $\sum_{n = 1}^{\infty}c_nf_n$ ...

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364 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) ...

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204 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} ...

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186 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 ...

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36 views

### bounds on a series with binomial coefficients

I have the following series
$\sum\limits_{l=1}^n {n\choose l} \alpha^{\beta^l}$
where $\alpha > 0$and $0 \leq \beta \leq 1$.
Can anybody guide me how I can evaluate it or find some tight upper ...

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236 views

### What is known about this series?

I recently came across the following function which intrigues me:
\begin{equation}
f(\alpha):=\sum_{i=0}^\infty \frac{\alpha^{i(i+1)/2}}{i!}.
\end{equation}
For $-1\leq \alpha\leq 1$ this function is ...

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205 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 ...

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95 views

### Finite series with reciprocal factorials

I asked the question at MSE
http://math.stackexchange.com/questions/982388/simple-finite-series-with-reciprocal-factorials
but got no answer or comment (it is not a homework).
I'm trying to find the ...

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**3**answers

368 views

### How did the summation operation come into use? [closed]

So we've been using summations at least since the dawn of calculus. I'm wondering how the process of summing a function came to be known? Are there events that led to the invention of the summation ...

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99 views

### The behavior of series involving special subsets of the prime numbers

It is well known that the series $\sum_{p\in \mathbb{P}} \frac{1}{p}$ diverges where $\mathbb{P}$ denotes the set of primes. Brun proved that $\sum_{p\in \mathbb{P_2}} \frac{1}{p}$ converges where $ ...

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### Inner Product of Given Sum Positive Sequence

Let $$A = \Big\{(a_1,a_2,\dots)\ \Big|\ a_i\ge 0, \sum_{i=1}^\infty a_i=1\Big\},$$ $$v(x)=\sup\left(\bigg\{\sum_{i=1}^\infty a_ib_i\ \bigg|\ (a_i)_{i=1}^\infty,\, (b_i)_{i=1}^\infty \in ...

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173 views

### Name for series $\sum f_n x^n / (n! (n+k)!)$

Let $(f_n)_{n\ge0}$ be a real sequence. Then $\sum f_n {x^n \over n!}$ is called the exponential generating function of $(f_n)$.
Let $k\ge0$ be a nonnegative integer. If we add another factorial ...

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230 views

### Is this series well known?

I recently encountered the following function
$$
f(t) = \sum_{n=0}^\infty \frac{t^{n^2}}{n^2!}.
$$
It seems familiar, though I cannot remember where I might have seen it before. I would like to know ...

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59 views

### Summation mollifier to ensure a certain alternating series has the correct value

I would like the function $f(n,M)$, where $n$ and $M$ are integers and $n\le M$, so that $f$ satisfies the following two conditions:
(1) $\sum_{n=0}^M (-1)^n f(n,M) = \tfrac{1}{2}$
(2) $f(n,M) ...

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### $\eta(s)$ expressed as an 'alternating' sum of Hurwitz Zetas. Why does it only work for sums with an even number of terms?

It is known that:
$$\zeta(s)= a^{-s}\,\sum_{k=1}^{a} \zeta_H\left(s,\frac{k}{a}\right)$$
is valid for all $a \in \mathbb{N}$ and all $s \in \mathbb{C}\,/1$, with $\zeta_H$ being the Hurwitz zeta ...

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### Conjectured relation between alternating Prime zeta series and Riemann zeta

Let $P(s)$ be the Prime zeta function.
Numerical evidence suggests these identities:
$$ \sum_{k=1}^\infty \frac{(-1)^{k}P(3k)}{k}=\log{\bigg(\frac{1}{945}\frac{\pi^6}{\zeta(3)}\bigg)}\qquad\quad ...

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### Algebraic Relations between $G(q)$ and $H(q)$

I had posted this originally on MSE, but got no response at all hence posting the same here.
In his paper "Algebraic Relations between Certain Infinite Products" (Proceedings of the London ...

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62 views

### Where the following interpolation method converges?

In this question about discrete-analytic functions (that is functions, who equal to their Newton series) I asked for a solution for the following problem:
Is there a method to extend the notion ...

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### If the generating function summation and zeta regularized sum of a divergent exist, do they always coincide?

One could assign a value to divergent series by means of several summation methods. One summation method we could consider is the generating function method. Let's sum, for example, the fibonacci ...

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244 views

### Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...

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140 views

### Without Skolem–Mahler–Lech Theorem? [closed]

Using Skolem–Mahler–Lech theorem one can easily prove the $\displaystyle \lim_{n\to +\infty}\left|\Re\left(\frac{1+i\sqrt{7}}{2} \right)^n\right| =+\infty$.
Is there a "simple way" to prove this ...

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161 views

### Do $r$-th root Harmonic numbers ever sum to integers?

None of the Harmonic numbers $H_n = \sum_{k=1}^n 1/k$ are integers for $n>1$
(e.g., this MSE question and answer).
Q. Define the $r$-th root Harmonic number $H_n^{1/r} = \sum_{k=1}^n ...

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131 views

### a question on sum of Gaussian binomial coefficients

I was trying to calculate something and at some point I get the following sum:
\begin{equation}
\sum_{t=0,t \text{ even}}^{s}{s+3n \brack s-t}\sum_{i = 0}^{t/2}q^{2i^2}{t/2+2n-i \brack t/2-i}{n ...

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145 views

### Proof of equidistribution theorem for exponential coefficients

Can anyone provide a proof of the equidistribution theorem using Weyl's criterion for the case of $c*a \,\,\, \text{(mod 1)}$ where $c=2^n: \,\,\, \forall n \in N_0$ for irrational algebraic $a$? The ...

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**1**answer

85 views

### Pointwise (in time) convergence in $H^{-1}$ implies pointwise weak convergence in $L^q$, why?

Let $u_n \to u$ in $C^0([0,T];H^{-1}(\Omega))$ and suppose $\lVert u_n \rVert_{L^\infty(0,T;L^\infty(\Omega))} \leq C$ for all $n$.
It follows that for almost all $t$, $u_n(t)$ is bounded in ...

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423 views

### A sum by Ramanujan for $\coth^{2}(5\pi)$

Ramanujan mentions in one of his letters to Hardy that $$\frac{1^{5}}{e^{2\pi} - 1}\cdot\frac{1}{2500 + 1^{4}} + \frac{2^{5}}{e^{4\pi} - 1}\cdot\frac{1}{2500 + 2^{4}} + \cdots = ...

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### closed form expression for an infinite series

Is there any closed form expression for the infinite sum $\sum_{n \geq 0}q^{n(n+1)/2}(1+q)(1+q^2)\cdots(1+q^n)u^n$ where both $q$ and $n$ are variables and $n \in N \cup {0}$?

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### a question about Tsirelson's space

NOTE: I asked this question over at math.stackexchange.com but got no answer or comments after 3 days, probably because it's a bit specialized. Hopefully it is interesting enough to ask over here.
...

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256 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 ...

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### A general question on nonnegative integer sequence [closed]

Let $A=\{x\ |\ x\in\mathbb Z_{\ge 0},\ x\ $ with some conditions$\ \}$.
Let $B=\mathbb Z_{\ge 0}-A$.
Define $\ 2A= \{a+b : a \in A,\ b \in A\}$.
Define $\ 2B=\{a+b : a \in B,\ b \in B\}$.
Then the set ...

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### Convergence and summation of power towers or Ackermann functions

I deleted this at StackExchange as it seems advanced or not very relevant for there and I want to ask here.
Let z be a complex number, $Im(z)\neq 0$ or $Re(z)<4$, m be a natural number, let ...

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### 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|} ...

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117 views

### How to classify the complex function with same natural boundary in complex plane? [closed]

There are complex functions with the same natural boundaries in the complex plane, but,they are different from each other. For example, there are lots of different lacunary power series with ...

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### If two functions are equal to their Newton series, is their composition also equal to its Newton series?

Suppose we have two real-valued functions $f(x)$ and $g(x)$, both equal to their Newton series expansion:
$$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k f\left (0\right)$$
$$g(x) = \sum_{k=0}^\infty ...

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### Computing $\int_0^T e^{itA}Be^{-itA} dt$ without an infinite series

I'm hoping to compute the following integral: $\int_0^T e^{itA}Be^{-itA} dt$ where $iA, iB$ are traceless anti-Hermitian matrices (i.e. $\mathfrak{su}(n)$). I have found the following form for the ...

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238 views

### On the sum of consecutive primes and product of first and last

Lets consider the sequence of consecutive prime numbers $p_1=2 , p_2=3 ,p_4=5 , \cdots$
. $(p_n,p_{j})$ is to be called good prime pair if $$\sum_{i =n }^{j}p_i= p_n p_{j}$$
Meaning the sum of set of ...

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92 views

### On the comparison of Egyptian fractions of two kinds

I posted the question on MSE here but it did not get any answer.
Consider $$S(n)=\left\{(a_1 ,a_2,a_3, \dots, a_n)\mid a_1\le a_2\le\cdots\le a_n, \; \sum_{r=1}^{n}\frac{1}{a_r} = 1\right\} \subset ...

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### About a failed conjecture on $d(N)$

This was post by me on Maths SE: but it did not get any solution
Some months ago I made the following conjecture -
Let $d(n)$ denote the number of divisors of $n $.
Then let $N$ be a number such ...

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**1**answer

280 views

### A question on null sequences

Is it true that a sequence of real numbers $\{a_n\}$ converges to zero if and only if the sequences $\{\sin^2(nh)a_n\}$ $(h \in \mathbb{R})$ all converge to zero?
In case the answer is ...

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363 views

### Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181:
3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79,
83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ...
Primes $p$ ...

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**1**answer

307 views

### Cesaro(?)/Euler(?) - summation of the $s(p)=\sum_{k=0}^\infty (-1)^{H(k)} (1+k)^p$ for $p=1,2,3,…$ (where $H(k)$ is the Hamming-weight)

In another thread (in MO) there was a question about a series where the signs at the terms alternate with the "Hamming-weight", that means according to the number of bits in the binary representation ...