For questions about recurrence relations, convergence tests, and identifying sequences

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96
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0answers
3k views

Identification of a curious function

During computation of some Shapley values (details below), I encountered the following function: $$ f\left(\sum_{k \geq 0} 2^{-p_k}\right) = \sum_{k \geq 0} \frac{1}{(p_k+1)\binom{p_k}{k}}, $$ where ...
25
votes
0answers
1k views

Does this sequence have any mathematical significance?

Take the sequence 001 and repeatedly append its second half to itself, using the larger half if the length is odd. This gives you ...
25
votes
0answers
882 views

When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like $$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$ and $$\infty ...
22
votes
0answers
270 views

Regular way to fill a $1\times1$ square with $\frac{1}{n}\times\frac{1}{n+1}$ rectangles?

The series $$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}=1$$ suggests it might be possible to tile a $1\times1$ square with nonrepeated rectangles of the form $\frac{1}{n}\times\frac{1}{n+1}$. Is there a ...
18
votes
0answers
405 views

Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$

In attempting to answer this question, I reduced it to a seemingly simple generating functions question, but after days of work was unable to construct a proof. Since I do not have experience trying ...
15
votes
0answers
245 views

Are the sums $\sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ transcendental?

This question is inspired by my answer to the question "How to compute $\prod_{n=1}^\infty\left(1+\frac{1}{n!}\right)$?". The sums $f(k) = \sum_{n=1}^{\infty} \frac{1}{(n!)^k}$ (for positive integer ...
15
votes
0answers
843 views

Quadratic Recurrence Relation

The following sequence appeared in IMC 2012 (a math competition): $$a_1 = \frac{1}{2}, \qquad a_{n+1} = \frac{n a_n^2}{1+(n+1)a_n}$$ I am trying to find an explicit formula for the sequence. It ...
13
votes
0answers
420 views

Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
12
votes
0answers
264 views

About the integral $\int_{0}^{+\infty}\sin(x\,\log x)\,dx$

It is an interesting exercise to show that the function $f(x)=\sin(x\log x)$ is Riemann-integrable over $\mathbb{R}^+$ (as shown by robjohn in this related question, for instance). Even more ...
12
votes
0answers
165 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
11
votes
0answers
115 views

Sum the series $\sum_{n = 0}^{\infty} (-1)^{n}\{(2n + 1)^{7}\cosh((2n + 1)\pi\sqrt{3}/2)\}^{-1}$

In one of his letters to G. H. Hardy, Ramanujan gave the following sum $$\dfrac{1}{1^{7}\cosh\left(\dfrac{\pi\sqrt{3}}{2}\right)} - \dfrac{1}{3^{7}\cosh\left(\dfrac{3\pi\sqrt{3}}{2}\right)} + ...
11
votes
0answers
271 views

Pólya and Szegő, Part I, Ch. 4, 174.

The following is a problem proposed in Pólya and Szegő's book "Problems and Theorems in Analysis" Assume that $0<f(x)<x$ and $$f(x)=x-ax^k+bx^\ell+x^\ell \varepsilon(x),\,\;\;\;\lim_{x\to ...
11
votes
0answers
365 views

Asymptotic related to the infinite product of sine

The amount is somewhat complicated ($x$ is a constant): $$S_n=\sum_{k=1}^n\ln\left(1-\frac{\sin^2\big(x/(2n+1)\big)}{\sin^2\big(k\pi/(2n+1)\big)}\right)\tag{*}$$ I want to enrich my handy powerful ...
11
votes
0answers
197 views

Function $f(x)$, such that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$

Consider a function $f(x)$. Define Taylor series $\sum_{n=0}^{\infty} f(n) x^n$. Is there such a function, other than constant $0$, that $\sum_{n=0}^{\infty} f(n) x^n = f(x)$? The Taylor series of ...
11
votes
0answers
333 views

Two questions about weakly convergent series related to $\sin(n^2)$ and Weyl's inequality

By using partial summation and Weyl's inequality, it is not hard to show that the series $\sum_{n\geq 1}\frac{\sin(n^2)}{n}$ is convergent. Is is true that ...
10
votes
0answers
202 views

A closed form for $\sum_{k=1}^\infty \psi^{(1)} (k+a)\psi^{(1)} (k+b)$

The following result $$ \sum_{k=1}^\infty\left(\psi^{(1)} (k)\right)^2 = 3\zeta(3) $$ where $\psi^{(1)}$ is the polygamma function makes me think there is a nice sum for the series $$ ...
10
votes
0answers
116 views

A sequence that avoids both arithmetic and geometric progressions

Sequences that avoid arithmetic progressions have been studied, e.g., "Sequences Containing No 3-Term Arithmetic Progressions," Janusz Dybizbański, 2012, journal link. I started to explore sequences ...
10
votes
0answers
233 views

The closed form of $\sum_{n=0}^{\infty} \arcsin\bigl(\frac{1}{e^n}\bigr)$

In my study on some type of integrals I met the series below that I don't how to approach it. Of course, one of the obvious questions is: does it have a closed form? Before answering that, I need to ...
10
votes
0answers
385 views

I can Euler-sum $\sqrt{-\ln(1)}-\sqrt{-\ln(2)}+\sqrt{-\ln(3)}-\cdots$. But how can I do $\sqrt{-\ln(1)}+\sqrt{-\ln(2)}+\sqrt{-\ln(3))}+\cdots$?

This is also related to an older thread in MSE ("what is the half derivative of zeta at zero?") . One of the possible steps in the problem of that thread was to evaluate the series ...
9
votes
0answers
62 views

Is there a simple expression for $\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$

Is there a simple expression for the power series $$\sum_{n=0}^{\infty} \left[ (4x)^n \frac{(n!)^2}{(2n+1)!} \right]^2?$$ This question came up in a quantum mechanics problem. Mathematica only returns ...
9
votes
0answers
278 views

A Ramanujan-like summation: is it correct? Is it extensible?

I'm still exercising with summation-procedures which I try to make correct Ramanujan-summations. Looking at the series $$ s(1/2,2) = (1/2)+(1/2)^4+(1/2)^9+(1/2)^{16}+... $$ and more general $$ s(b,p) ...
9
votes
0answers
250 views

To how many decimals is $\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$ correct?

Consider: $$\sum_ {k=1}^\infty \frac{k}{\sqrt{k!}} = \frac{49850839\,\pi}{29567947}$$ This is, as far as I'm able to check with my software, correct to at least 167 decimals. If anyone has the ...
8
votes
0answers
98 views

Inequality involving exponential partial sums

Consider the exponential partial sums $E_n(x) = \sum_{i=0}^n \frac{x^i}{i!}$. I want to prove that for all $x \ge 0$: $$2 \frac {E_{n-1}(x)} {E_n(x)} \ge \frac {E_{n}(x)} {E_{n+1}(x)} + \frac ...
8
votes
0answers
75 views

Does $\sum_n |\sin n|^{cn^2}$ converge?

So I recently asked a question about convergence of $\sum_n |\sin n|^{cn}$ for arbitrary $c > 0$ and it turns out that the terms of the series don't even converge, for any $c > 0$, so the series ...
8
votes
0answers
216 views

How to prove there exists $n_{1}a_{n_{0}}+n_{2}a_{n_{1}}+\cdots+n_{k}a_{n_{k-1}}<3(a_{1}+a_{2}+\cdots+a_{N})$

Let $a_{1},a_{2},\cdots,a_{N}$ be nonnegative reals, not all $0$. Prove that there exists a sequence $$1=n_{0}<n_{1}<\cdots<n_{k}=N+1$$ of integers such that ...
8
votes
0answers
115 views

A cotangent series related to the zeta function

$$\sin x = x\prod_{n=1}^\infty \left[1-\frac{x^2}{n^2\pi^2}\right]$$ If you apply $\log$ to both sides and derivate: $$\cot x = \frac{1}{x} - \sum_{n=1}^\infty \left[\frac{2x}{n^2\pi^2} ...
8
votes
0answers
192 views

How find this series $\sum_{n=1}^{\infty}\frac{1}{n^2H_{n}}$?

Question: This follow series have simple closed form? $$\sum_{n=1}^{\infty}\dfrac{1}{n^2H_{n}}$$ where $$H_{n}=1+\dfrac{1}{2}+\dfrac{1}{3}+\cdots+\dfrac{1}{n}$$ I suddenly thought of this ...
8
votes
0answers
157 views

How to prove that my sum coincides with a sequence on the Online Encyclopedia of Integer Sequences?

I have a topological space $X$ whose reduced $\bmod 2$ Betti numbers (that is to say, the dimension of the $\bmod 2$ reduced homology) I computed to be $$\small \dim \tilde{H}_t(X; {\mathbb{Z}}_2) = ...
8
votes
0answers
207 views

Invariant functions on the space of finite sequences of reals

Let $S$ be a space of all finite sequences of real numbers (we don't endow it with metric or topology in general). Before asking the main question, some notation. 1. For each $\mathbf s\in S$ we ...
7
votes
0answers
168 views

Does $A193201$ count the partitions of $n$ of arbitrary dimension?

By my count, these sequences match for $n=0\ldots6$, where partitions that are the same after relabeling dimensions are considered equivalent (i.e., the dimensions are unordered). For example, for ...
7
votes
0answers
146 views

Closed form of $\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$

Find the closed form of $$\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$$ I know the closed form for smaller powers like $2, 3$ exists, but I'm not sure if there is a closed form for this ...
7
votes
0answers
124 views

Another way of expressing $\sum_{k=0}^{n} (-1)^k\frac{H_{k+1}}{n-k+1}$

In this post Another way of expressing $\sum_{k=0}^{n} \frac{H_{k+1}}{n-k+1}$ I asked for a solution of the non-alternating series. How about the alternating series? Can we find a nice way of ...
7
votes
0answers
173 views

Continued fraction and double series.

From Euler's continued fraction formula, we have $$x = \cfrac{1}{1 - \cfrac{r_1}{1 + r_1 - \cfrac{r_2}{1 + r_2 - \cfrac{r_3}{1 + r_3 - \ddots}}}}\,$$ and $$x = 1 + \sum_{i=1}^\infty r_1r_2\cdots r_i = ...
7
votes
0answers
103 views

Is this a recurrence for the characteristic sequence of composite numbers?

The characteristic sequence of composite numbers is equal to 1 if $n$ is not a prime number and equal to 0 if $n$ is a prime number, starting: $$1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1,...$$ where the ...
7
votes
0answers
320 views

Extension of the Jacobi triple product identity

The Jacobi triple product identity is: $$\prod\limits_{n=1}^{ \infty }(1-q^{2n})(1+zq^{2n-1})(1+z^{-1}q^{2n-1})=\sum\limits_{n = - \infty }^ \infty z^n q^{n^2} $$ I would like to extend the idea for ...
7
votes
0answers
426 views

On the weak and strong convergence of an iterative sequence

I have some difficulties in the following problem. I would like to thank for all kind help and construction. Let $H$ be an infinite dimensional real Hilbert space and $F: H\rightarrow H$ be a ...
7
votes
0answers
240 views

A sequence similar to the Catalan numbers

The $n$-th Catalan number $c_n$ has the closed form $\frac1{n+1}\binom{2n}{n}$ and follows the recursion $c_n = \sum\limits_{i = 0}^{n-1} c_{n-1-i}c_i$ I am interested in the quantity $e_n$ which ...
6
votes
0answers
155 views

How to sum up this series and simplify yet another one?

Primarily, I would like to know what could be done with this series: $$ \sum_{n=2}^{\infty}\frac{n^3}{(n^2-1)^3}\left(\frac{n-1}{n+1}\right)^{2n}$$ As hardmath says in his comment, the series ...
6
votes
0answers
103 views

Minimizing beginning terms of a Fibonacci-like sequence

For $ a, b \in \mathbb{Z}^+ $ such that $ a = b $ or $ 2a < b $, let $ f_{a, b} : \mathbb{Z}^+ \to \mathbb{Z}^+ $ be a sequence that satisfies the following three criteria: $$ f(1) = a, \\ f(2) = ...
6
votes
0answers
94 views

Name of a certain set

I want to know if there is any already-standard way to refer to the sets described as follows. For a set $X$, let $-X = \{-x: x \in X \}$; call it the negative of $X$. Take the set of all primes in ...
6
votes
0answers
117 views

Visual Proofs of Series Summations

I'd like to put together a compilation of visually geometric proofs of series summations. I have three famous 2D examples to clarify what I mean below, but other "visually geometric" proofs of an ...
6
votes
0answers
86 views

Evaluating a series involving inverse of $\sinh(n\pi)$

$$-\frac{8\varepsilon_0V_0}{\pi}\sum_{n \text{ odd}}\frac{1}{n\sinh(n\pi)}=\boxed{\displaystyle-\frac{\varepsilon_0V_0}{\pi}\ln 2.}$$ I have not found a way to sum this series analytically. ...
6
votes
0answers
396 views

Convergence/Divergence of infinite series $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\left|{\cos n}\right|}}$

It is well known that $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$ is divergent while $ \displaystyle\sum_{n=1}^{\infty} \frac{1}{n^{1+\epsilon}}$ is convergent for a fixed positive value of ...
6
votes
0answers
250 views

expansion for $1-|t|$

Let $f$ be a continuous function on $\mathbb{R}$ with compact support with exactly one maximum. Form the functions $$ f_{m,k}(x)=f^m\left(x-\frac{k}{2^m}\right) $$ I am wondering if one can expand ...
5
votes
0answers
51 views

Does such a series exist?

This question came up as some puzzle. Does there exist a sequence of real numbers ${c_j}$ such that $\sum{c_j^m} = m$ for all positive integers $m$? I argue no. Suppose there exists such a ...
5
votes
0answers
65 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
5
votes
0answers
95 views

Infinite series involving factorials of squares

Does $$\sum_{n=0}^\infty \frac{1}{(n^2)!}=2.04167\dots$$ possess a closed form?
5
votes
0answers
48 views

Calculus II: Comparison Test

I have this math problem where I have to show that a sum converges. Is this correct? Thanks $$\sum_{n=1}^{\infty}\frac{2n-1}{ne^n}$$ I chose $\sum_{n=1}^{\infty}\frac{2n}{ne^n}$ to compare it to. ...
5
votes
0answers
83 views

Find the closed form of the digamma related series

The question I asked here Computing $\sum_{n=1}^{\infty} \left(\psi^{(0)}\left(\frac{1+n}{2}\right)-\psi^{(0)}\left(\frac{n}{2}\right)-\frac{1}{n}\right)$ made me think to ask for your support for ...
5
votes
0answers
132 views

Sum of infinite series defined recursively

Suppose $s$, $t$, and $\delta$ are constants satisfying $0<s<t<1$ and $\delta>1$. An infinite sequence $\{y_k\}_{k=1}^{\infty}$ defined as follows: The initial term, $y_1$, is the ...