Questions tagged [sequences-and-series]

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Is this long closed form for pi trivial?

With the help of wolfram alpha we got very long closed form for $\pi$ in terms of algebraic numbers, logarithms of algebraic numbers and $cot^{-1},coth^{-1}$ which can be expressed as logarithms. From ...
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7 votes
3 answers
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Why mpmath computes $\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right)$

Working with precision 500 decimal digits, mpmath in sage computes: $$\sum_{n=2}^\infty (-1)^n\log(n)=\log\left(\frac1 2 \sqrt{2} \sqrt{\pi}\right).\tag{1}\label{1}$$ We believe the LHS of \eqref{1} ...
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3 votes
3 answers
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Surprisingly long closed form for simple series

For natural $A$ define $$ f(A)=\sum_{n=1}^\infty \frac{1}{A^n}\left(\frac{1}{An+1}- \frac{1}{An+A-1}\right)$$ $f(A)$ is BBP (Bailey-Borwein-Plouffe) formula and allows digit extraction in base $A$. ...
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3 votes
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Examples of infinite dimensional involutions

Examples of infinite dimensional involutions I'm looking for more examples of involutions of the type portrayed below, in which two sets of indeterminates (real or complex) each can be transformed ...
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2 votes
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49 views

The Laplace transform and the Lagrange compositional inversion formula

I'm looking for references which derive the Lagrange inversion formula, given below (in bold), for the Taylor series coefficients of the compositional inverse of a function $f$ analytic at the origin ...
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Given the formula for each specific term of a sequence how do you find the sum of n terms of the sequence? An = 3*(-3)^n-1. Find sum n = 1 to n = 5? [closed]

Which summation properties/rules are used in determining the summation of a finite series ? The given formula an = 3* (-3)^n-1 gives the value for each specific n term, but how to determine the sum ...
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2 votes
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233 views

What's the fastest way to compute $\log n$ for $n>1$?

As it is well known, if $|x|<1$ then we can compute $\log(1+x)$ by the Taylor series $$\log(1+x)=x-\frac{x^2}2+\frac{x^3}3-\cdots.$$ Thus, to compute $\log n$ with $n>1$, we may employ the ...
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-1 votes
0 answers
245 views

$f' = e^{f^{-1}}$, a third time

I am of the impression the differential equation $f' = e^{f^{-1}}$ was considered on mathoverflow for the first time here: How do i solve this : $\displaystyle \ f'=e^{{f}^{-1}}$? It was found ...
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1 answer
79 views

Sequence of reals such that $x_{n+1}\leq ab^{n}x_{n}^{1+s}$ converges to $0$?

Let $\{x_{n}\}_{n=0}^{\infty}$ be decreasing sequence of non-negative reals. Suppose that there exist constants $a, s>0$ and $b>1$ such that $$x_{n+1}\leq ab^{n}x_{n}^{1+s}$$ and $$x_{0}\leq a^{-...
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Is there a residue sum formula in quaternionic analysis?

In complex analysis, there is a formula involving the residues of complex functions that one can employ to find the value of certain infinite series. If the function $f: \mathbb{C} \to \mathbb{C} $ ...
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1 answer
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A conjectural identity involving infinite series

Recently I formulated the following curious conjecture based on my computation. Conjecture. For all $|x|>1$, we have the identity $$\sum_{k=0}^\infty\frac{\sum_{j=0}^{k}\binom{2k+1}{2j}(1-x)^jx^{k-...
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3 votes
1 answer
211 views

One series converges iff the other converges

In Show that $\sum\limits_pa_p$ converges iff $\sum\limits_{n}\frac{a_n}{\log n}$ converges it is said that this sequence of partial sums converges $$ \begin{split} \sum_{1<n\leq N}\frac{a_{n}}{\...
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160 views

$\lim_{x\to \infty} \left(\sum_{n\leq x} (\log n)^k/n - \int_1^x (\log t)^k/t\right) = \text{?}$

It is easy to see (by Euler-Maclaurin, say, or just by thinking of a graph) that $$\lim_{x\to \infty} \sum_{n\leq x} \frac{(\log n)^k}{n} = \int_1^x \frac{(\log t)^k}{t} + C + O\left(\frac{(\log x)^k}{...
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4 votes
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174 views

Karamata's Abelian/Tauberian Theorem in the complex plane

The following result is well known (a particular case of Karamata's Tauberian Theorem for Power Series in Corollary 1.7.3 of Regular variation by Bingham, Goldie & Teugels): Fix $c, \rho>0$. If ...
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1 vote
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100 views

Self-referencing recurrence relation with exponential

I have the self-referencing recurrence relation $$ d(0) = 0 $$ $$ d(1) = a $$ $$ d(n+1) = d(n) + a*e^{-(\frac{d(n)-n*c}{b})^4} $$ Written as a sum: $$ d(N) = a*\sum_{n=0}^{N}e^{-(\frac{d(n)-n*c}{b})^...
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2 votes
1 answer
131 views

Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?

It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
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0 answers
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Evaluations of two new series involving Lucas $v$-sequences

Let $A$ and $B$ be integers. The Lucas $v$-sequence $v_n(A,B)\ (n=0,1,2,\ldots)$ is defined by $v_0(A,B)=2,\ v_1(A,B)=A$, and $$v_{n+1}(A,B)=Av_n(A,B)-Bv_{n-1}(A,B)\ \ \ (n=1,2,3,\ldots).$$ From the ...
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6 votes
0 answers
64 views

q-binomial-like series with exponentials defining probability distribution

Recently I encountered the series $$f(d) = \frac{1}{(t;t)_\infty} \sum_{k=0}^\infty \frac{(-1)^k t^{\binom{k}{2}}}{(t;t)_k} e^{-t^{d-k}}$$ where $(t;t)_n = \prod_{i=1}^n (1-t^i)$, and $0 < t < 1$...
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1 vote
1 answer
107 views

Numbers $m$ for which coefficients of the polynomial $p(m,x)$ are relatively prime

From A248667: The polynomial $p(n,x)$ is defined as the numerator when the sum $$1 + \frac{1}{nx + 1} + \frac{1}{(nx + 1)(nx + 2)} + \cdots + \frac{1}{(nx + 1)(nx + 2)\cdots(nx + n - 1)}$$ is written ...
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0 votes
1 answer
143 views

Number of positive integers $k$ such that there exists a nonnegative integer $m$ with $k + k^m = n$

Let $a(n)$ be the number of positive integers $k$ such that there exists a nonnegative integer $m$ with $k + k^m = n$. The sequence begins $$0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3$$...
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0 answers
327 views

Is $\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi$?

It seems that $$\lim_{x\to\infty}\sum_{n=-\infty}^{\infty}\frac{x}{n^2+x^2}=\pi.$$ But I can't prove it. I cannot prove that the function is decreasing in $x$ either.
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1 vote
0 answers
59 views

Geometric series involving the Laguerre polynomials

Let put $\alpha=5$ and $x=3$. Consider the following set given by $$M=\lbrace \; n \in N, \; \; 0 < |L_{n}^{5}(3)| < 1 \; \rbrace$$ Where $L_{n}^{\alpha}(x)$ is the generalized Laguerre ...
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3 votes
1 answer
76 views

Tauberian lower bound for a series

Let $(a_n)_{n\in\mathbb{N}}$ be a sequence of positive number such that $\sum_n a_n < +\infty$ (i.e. $a_n \in \ell^1$) but $\sum_n r^n a_n = +\infty$ for every $r > 1$. Given $\sigma \in (0,1)$, ...
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2 votes
1 answer
188 views

Linear combinations of geometric series

Consider the uncountable-dimensional vector space $V$ consisting of finite linear combinations of infinite sequences of the form $(1,z,z^2,z^3,\dots)$ with $z \neq 1$ in $\mathbb{C}$. Since the ...
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9 votes
2 answers
269 views

Asymptotics of a quadratic recursion

Consider the sequence defined by \begin{align} c_0 &{}= 1 \\ c_n &{}= 2\,n\,c_{n-1}-\frac{1}{2}\sum_{m=1}^{n-1}c_m\,c_{n-m}. \end{align} How can you prove that it has the following asymptotics ...
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1 vote
1 answer
84 views

Upper bound on double series

We consider the sum $$ \sum_{m \in \mathbb Z^2} \frac{1}{(3 m_1^2+3m_2^2+3(m_1+m_1m_2+m_2)+1)^2}. $$ Numerically, it is not particularly hard to see that the value of this series is well below $4$, ...
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2 votes
1 answer
122 views

As-closed-as-possible formula for an integral and/or sum

I need to find the solution of this integral: $$\int^\pi_{-\pi}{d\varphi\,{}\frac{\Gamma(n,2\pi\varphi)}{(1+ia\varphi)^n}},\tag{1}\label{1}$$ where $a\in(0,1)$ and $n$ is a positive integer (not zero)....
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6 votes
3 answers
513 views

Series solution for general trinomial

Consider the equation $x^5-2x^2+z=0$ How do you derive the Lagrange inversion theorem series solution for it? I know it exists because the answer is here for any trinomial https://arxiv.org/pdf/0910....
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65 votes
2 answers
3k views

Function that produces primes

For any $n\geq 2$ consider the recursion \begin{align*} a(0,n)&=n;\\ a(m,n)&=a(m-1,n)+\operatorname{gcd}(a(m-1,n),n-m),\qquad m\geq 1. \end{align*} I conjecture that $a(n-1,n)$ is always ...
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7 votes
1 answer
493 views

Infinite series for $1/\pi$. Is it known?

Indirect method (associated with a certain problem of electrostatics) indicates that $$\sum\limits_{j=1}^\infty \frac{(2j-3)!!\,(2j-1)!!}{(2j-2)!!\,(2j+2)!!}=\frac{2}{3\pi}.$$ Is this result known?
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0 votes
0 answers
87 views

Conjecture on a sieve of Flavius Josephus

Flavius Josephus's sieve: Start with the natural numbers; at the $k$-th sieving step, remove every $(k+1)$-st term of the sequence remaining after the $(k-1)$-st sieving step; iterate. Some examples: ...
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1 vote
1 answer
55 views

Stern-Brocot tree and subtree

Let $a(n)$ be A007306, denominators of Farey tree fractions (i.e., the Stern-Brocot subtree in the range $[0,1]$). Let $b(n)$ be A002487, Stern's diatomic series (or Stern-Brocot sequence): $b(0) = 0, ...
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2 votes
1 answer
148 views

Another combinatorial identity

Is it true that $$\sum_{r=0}^p \sum_{i=0}^r a_{n,p,r,i}=0$$ for all natural $n$ and all natural $p\ge2n$, where $$a_{n,p,r,i}:=\frac{(-1)^r (n+p-r-1)! (n p-i (r-i))}{i!(r-i)! (n-i)! (p-r+i)! (n-r+i)! ...
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4 votes
3 answers
195 views

Infinite sum of Laguerre polynomials: is $\sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^kL_n^k(x)^2=e^x+P(x)$, with $P$ a polynomial of degree $2n-1$?

I have the following expression: $$ \sum_{k=0}^{\infty}\frac{n!}{(k+n)!}x^k(L_n^k(x))^2, $$ where $$ L_n^k(x)=\sum_{j=0}^n(-1)^j\binom{n+k}{n-j}\frac{x^j}{j!} $$ is the usual associated Laguerre ...
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1 vote
1 answer
120 views

Connection between central factorial numbers and the Stern–Brocot tree

Consider the central factorial numbers of even indices formed by $$U(n,k)=\frac1{(2k)!}\sum_{i=0}^{2k}(-1)^i\binom{2k}i(k-i)^{2n}.$$ Let $u(n,k):=U(n,k)\mod 2$. Define the triangle of numbers $$A(r,j)=...
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3 votes
1 answer
182 views

Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$

Note: Posting in MO since it was unanswered in MSE Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...
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2 votes
0 answers
88 views

Equality of bivariate formal series

Is it possible to prove algebraically that the two series uniquely defined by the following equations are equal: $L_1=uz+zL_1^2+z \partial_uL_1$ and $L_2=uz+z^2+z L_2^2+2z^4 \partial_zL_2$
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0 votes
0 answers
68 views

$\max\left\lbrace(n-i+1)\operatorname{prime}(i); 1 \leqslant i \leqslant n\right\rbrace$ from permutation

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and sorting in descending order if $n$ is prime sorting in ascending order if $n$ is not prime every $n$ ...
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3 votes
0 answers
196 views

On the density of a particular subset of integers

Given a positive integer $n$ in the standard form $$n=\prod_k p_k^{\alpha_k}$$ and the arithmetic function (investigated by Erdős in this paper) $$A(n)=\sum_k \alpha_k p_k$$ let's define the subset $E$...
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2 votes
1 answer
179 views

Cauchy's integral formula and essential singularities

Let $f$ be holomorphic at $z_0\in\mathbb C$. I would like to compute the integral $$\oint_{\gamma_{z_0}} f(z)\, e^{\frac{1}{z-z_0}}dz,$$ where $\gamma_{z_0}$ is a small circle around $z_0$. By ...
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1 vote
0 answers
73 views

Question regarding convergent series of positive real numbers [closed]

If we have two convergent series of positive reals, $∑b_n$ and $∑c_n$, can we find a third convergent series of positive reals, $∑a_n$ , such that $\frac{a_n}{b_n }$ $\rightarrow$ $\infty$ and $\frac{...
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0 votes
0 answers
79 views

Prime numbers from another permutation

Related question: Prime numbers from permutation Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n-1)$ fixed and forward-cyclically permuting every $n$ consecutive ...
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8 votes
4 answers
271 views

How to find the asymptotics of a linear two-dimensional recurrence relation

Let $d$ be a positive number. There is a two dimensional recurrence relation as follow: $$R(n,m) = R(n-1,m-1) + R(n,m-d)$$ where $R(0,m) = 1$ and $R(n,0) = R(n,1) = \cdots = R(n, d-1) = 1$ for all $n,...
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6 votes
2 answers
771 views

Has the "partial Sophomore's Dream function" been studied before?

We can consider the generalized Harmonic numbers $$H_{n,m} := \sum_{k=1}^{n} \frac{1}{k^{m}} $$ as a partial version of the Riemann zeta function, because $$\lim_{n \to \infty} H_{n,m} = \zeta(m). $$ ...
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9 votes
1 answer
482 views

Prime numbers from permutation

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n)$ fixed and reverse-cyclically permuting every $n$ consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $...
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2 votes
0 answers
176 views

Conjecture on A057030

Let $P(n)$ of a sequence $s(1),s(2),s(3),...$ be obtained by leaving $s(1),...,s(n-1)$ fixed and reversing every n consecutive terms thereafter; apply $P(2)$ to $1,2,3,...$ to get $PS(2)$, then apply $...
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0 votes
1 answer
293 views

The exploration of the asymptotic behavior of a simple sum. $\sum_{k=1}^{\infty} (k^{1/k} - 1)$ [closed]

An informal investigation of a sum. Consider this sum: $$S =\sum_{k=2}^{\infty}(k^{1/k} -1)$$ Does this converge? How does it behave as it diverges, if it diverges? If $k$ equaled $1$ we would get $0$ ...
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0 votes
1 answer
85 views

Property of composite numbers

Let $a(n)$ be the sequence of composite numbers (starting from $4$). Let $$b(n)=a(n-1)a(n-2) \operatorname{mod} a(n)$$ Obviously, $b(1)=b(2)=0$. I conjecture that with the only exception for the $b(3)=...
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1 vote
0 answers
98 views

Questions about iterating the Euler-Maclaurin summation formula

Introduction The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
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  • 3,463
1 vote
1 answer
232 views

A discrete version of Poincaré's inequality

Given a (bounded) sequence $\{q_n\}_{n\geq 0}$ such that $\lvert q_n\rvert \leq 1$ for all $n \geq 0$ and $\sum_{n\geq 0} q_n = 0$. We can impose the condition that $\sum_{n\geq 0} \lvert q_n\rvert \...
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