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

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17 views

### Generating sequence [on hold]

I was trying to generate a sequence of numbers.
Let it be (i j k),
...

**2**

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**5**answers

328 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 ...

**0**

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**0**answers

84 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

**1**answer

361 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 ...

**1**

vote

**2**answers

357 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, ...

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**0**answers

89 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
...

**1**

vote

**1**answer

105 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|} ...

**5**

votes

**2**answers

206 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 ...

**11**

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**4**answers

462 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

**1**answer

806 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) ...

**24**

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**3**answers

738 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,
...

**2**

votes

**2**answers

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} ...

**27**

votes

**1**answer

441 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) = ...

**30**

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**3**answers

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 ...

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votes

**6**answers

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 ...

**3**

votes

**1**answer

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 ...

**0**

votes

**1**answer

111 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 ...

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**3**answers

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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**1**answer

645 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 ...

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**0**answers

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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votes

**1**answer

236 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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**1**answer

187 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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**2**answers

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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**0**answers

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 ...

**2**

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**2**answers

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

369 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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**0**answers

100 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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**2**answers

85 views

### 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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**1**answer

488 views

### What was Lambert's solution to $x^m+x=q$?

I've been thinking about Lambert's Trinomial Equation quite a bit, and I want to see his solution. The only solution I could find was in Euler's form, and I still don't quite understand how he got ...

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**4**answers

804 views

### Two Equal Series?

Hey all,
I'd like to know if anyone has a link to a proof of the following?
Take two infinite sequences $a_n$ and $b_n$ such that
$$\sum_{n=1}^\infty a_n^s=\sum_{n=1}^\infty b_n^s=finite$$
for all ...

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**1**answer

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 ...

**1**

vote

**1**answer

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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**2**answers

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) ...

**0**

votes

**0**answers

52 views

### $\eta(s)$ 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 ...

**2**

votes

**2**answers

326 views

### 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 ...

**3**

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**2**answers

362 views

### Sum of series $a^{i^2}$

Is there any closed form known for the expression $\sum_{i=1}^\infty a^{i^2}$ where $|a|<1$? Thanks!

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**0**answers

47 views

### 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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**3**answers

1k views

### Convergent subsequence of $\sin n$

It is well known (not to me -- ed.) that for every real number $\theta \in [0, 1]$ there exists a sequence $(k_i)$ such that $\lim\sin k_i = \theta,$ but there appear to be no explicit such ...

**0**

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**0**answers

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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**0**answers

156 views

### 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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**1**answer

246 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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**0**answers

141 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 ...

**4**

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**1**answer

381 views

### An infinite set of identities using Stirling numbers 1st kind - are they all zero?

I have the following set of series involving the Stirling numbers 1'st kind and binomials, which can be understood as a set of dot-products of row- and column-vectors of two infinite matrices (where R ...

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**0**answers

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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**9**answers

5k views

### What is the indefinite sum of tan(x)?

What is the indefinite sum of the tangent function, that is, the function $T$ for which
$\Delta_x T = T(x + 1) - T(x) = \tan(x)$
Of course, there are infinitely many answers, who all differ by a ...

**0**

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**0**answers

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 ...

**0**

votes

**2**answers

146 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 ...

**0**

votes

**1**answer

86 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 ...

**20**

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**4**answers

1k views

### Freeness of a Z[x]-module

Definition: Call a mapping $f: \mathbb{Z} \rightarrow \mathbb{Z}$
a generalized polynomial if for any distinct integers $m$ and $n$
we have $(m - n)|(f(m)-f(n))$.
It is easy to check that polynomial ...

**14**

votes

**1**answer

424 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 = ...