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23 votes
4 answers
2k views

Identity for an infinite product

Here is an experimental "result" exhibiting the difference of two (formal) infinite products that "almost factorizes". QUESTION. Is this true? $$\prod_{n\geq1}(1+x^{2n-1})^{24} - \...
T. Amdeberhan's user avatar
15 votes
4 answers
3k views

No Tonelli or Fubini

Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
15 votes
2 answers
473 views

Generalizations of summation methods of divergence series

If one looks at the "summation proofs" of divergent series such as Grandi's series, one might see a pattern that most of the computation rely on linearity and comparability with the shift ...
Serge the Toaster's user avatar
13 votes
3 answers
810 views

Is $\sum_{n=1}^\infty\frac{S(n)}{n!}$ an irrational, where $S(n)$ denotes the sum of remainders function?

For each integer $n\geq 1$ we consider the arithmetic function $$S(n)=\sum_{k=1}^n n\text{ mod }k,\tag{1}$$ the sum of remainders function, the arithmetic function A004125 from the OEIS. Example. We'...
user142929's user avatar
11 votes
2 answers
1k views

Two divergent series conspiring?

Consider the sequence $a_n=2^{2n}\binom{2n}n^{-1}$. Stirling's approximation shows that $a_n\sim \sqrt{\pi n}$, thus $$\sum_{n\geq0}\frac{\pi}{2a_n}\qquad \text{and} \qquad \sum_{n\geq0}\frac{a_n}{2n+...
T. Amdeberhan's user avatar
9 votes
2 answers
354 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 ...
Matteo Beccaria's user avatar
7 votes
1 answer
268 views

A differential equation governing compositional inversion

Looking for references for the following theorem. Given the formal Taylor series/exponential generating function $$T(z) = \sum_{n \ge 1} a_n \; \frac{z^n}{n!},$$ for which the indeterminates $a_n$ and ...
Tom Copeland's user avatar
  • 10.5k
7 votes
1 answer
1k views

Signed variant of the Flint Hills series

I asked my Calculus 2 students to come up with a series the convergence of which they are unable to decide. One of the students, Denis Zelent, invented a very interesting one: $$ \sum_{n = 1}^\infty \...
Mateusz Kwaśnicki's user avatar
6 votes
1 answer
234 views

What about of periodic points of $\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n$, $0<x<1$, where $\mu(n)$ is the Möbius function?

Let $\mu(n)$ the Möbius function, we define $F:[0,1]\to[0,1]$ as $$F(x)=\sum_{n=1}^\infty\frac{\mu(n)}{n}x^n.\tag{1}$$ For a function of this kind (I presume that this continuous function has image $[...
user avatar
6 votes
3 answers
536 views

A need for analytic continuation of a finite sum function

Let $\varphi(n):=(-1)^{n+1}(n+1)2^{2n}$. I am able to prove the following identity (${\color{red}{\mathbf{LHS}}}$=infinite series, ${\color{blue}{\mathbf{RHS}}}$=finite sum) \begin{align*} {\color{red}...
T. Amdeberhan's user avatar
5 votes
1 answer
279 views

Is there a theorem which provides conditions under which a power series satisfies the reciprocal root sum law?

Kalman - Six ways to sum a series discusses Euler's original proof for the Basel problem $\sum\limits_{n=1}^\infty \frac{1}{n^2}=\frac{\pi^2}{6} $: $$\frac{\sin(\sqrt x)}{\sqrt x} = 1- \frac{x}{3!}+ \...
pie's user avatar
  • 541
5 votes
1 answer
680 views

When does this interesting sum diverge?

For $x \gt 0,$ what is the greatest $y$ such that $$\sum_ {1\le h^x \le k^y} \frac{1}{h^x k^y}= \infty ?$$ I don't know of any references or methods for this -- not even for $x=1$, for which the ...
Clark Kimberling's user avatar
5 votes
1 answer
882 views

Two integral representations for $\zeta(3)$ from Zurab's integral and standard formulas for the gamma function

This morning I wrote with the help of a CAS, and integral representation for the Apéry's constant $\zeta(3)$ and some standard formulas two formulas involving this constant. I would like to know if ...
user avatar
5 votes
2 answers
1k views

Summation of double exponential series

Let $q \in (0,1)$ and consider the following summation: $$S(q,n) = \sum_{i=1}^n {q^2}^i$$ Is there a closed form expression or upper and lower bounds for $S(q,n)$? Specifically, I am looking for ...
Kcafe's user avatar
  • 519
4 votes
2 answers
592 views

From Zurab's integral representation for the Apéry's constant to almost impossible integrals

I would like to know if the following integrals are known, or in case that aren't in the literature we can calculate these in closed-form (in terms of elementary and standard functions). I wondered ...
user142929's user avatar
3 votes
1 answer
631 views

Is the sequence $(\log(n!)\mod1)_{n\in\mathbb N}$ dense in the interval $[0,1]$?

This question was raised in the comment by Todd Trimble at how to proof there is a natural number n, the first four digits of n! Is 2018?. I thought the question may be posted separately, as even ...
Iosif Pinelis's user avatar
3 votes
1 answer
334 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 $c>...
Anand's user avatar
  • 1,649
3 votes
1 answer
128 views

Weaker version of the lemma of K.L. Chung

Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds: $$...
giorgi nguyen's user avatar
2 votes
2 answers
152 views

Divergence rate of geometric sum of random variables

Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of strictly positive and identically distributed random variables and let $\beta\le 1$. I am trying to prove that $$ 0<\lim_{\beta\rightarrow 1}(1-\...
Marc's user avatar
  • 479
2 votes
0 answers
161 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 ...
Tom Copeland's user avatar
  • 10.5k
1 vote
1 answer
457 views

A (surprising?) expression for $e$

I apologise if this is off topic. Consider the quantity $$ F(m,n,k)=\frac{(m)_k}{k!n^{k-1} } $$ where $m,n \in \mathbb{N}.$ For moderately large $n$, it seems that the approximation $$ \sum_{k=1}^{K} ...
kodlu's user avatar
  • 10.4k
1 vote
0 answers
162 views

Triangular and pentagonal numbers in $q$-series

Consider the following two infinite series $$\sum_{n\geq0}a(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{5k})^2} \,\,\,\, \text{and} \,\,\, \sum_{n\geq0}b(n)q^n=\prod_{k\geq1}\frac1{(1-q^k)^2(1-q^{7k})^2}...
T. Amdeberhan's user avatar