Questions tagged [sequences-and-series]

for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.

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166 votes
3 answers
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Convergence of $\sum(n^3\sin^2n)^{-1}$

I saw a while ago in a book by Clifford Pickover, that whether the Flint Hills series $\displaystyle \sum_{n=1}^\infty\frac1{n^3\sin^2 n}$ converges is open. I would think that the question of its ...
Andrés E. Caicedo's user avatar
113 votes
4 answers
25k views

Is the series $\sum_n|\sin n|^n/n$ convergent?

Problem. Is the series $$\sum_{n=1}^\infty\frac{|\sin(n)|^n}n$$convergent? (The problem was posed on 22.06.2017 by Ph D students of H.Steinhaus Center of Wroclaw Polytechnica. The promised prize for ...
Lviv Scottish Book's user avatar
100 votes
8 answers
12k views

Is $ \sum\limits_{n=0}^\infty x^n / \sqrt{n!} $ positive?

Is $$ \sum_{n=0}^\infty {x^n \over \sqrt{n!}} > 0 $$ for all real $x$? (I think it is.) If so, how would one prove this? (To confirm: This is the power series for $e^x$, except with the ...
J Russell's user avatar
  • 1,103
86 votes
4 answers
10k views

Nonexistence of boundary between convergent and divergent series?

The following is a FAQ that I sometimes get asked, and it occurred to me that I do not have an answer that I am completely satisfied with. In Rudin's Principles of Mathematical Analysis, following ...
Timothy Chow's user avatar
75 votes
1 answer
15k views

A number theory problem where pi appears surprisingly

For a given positive integer $M$, the sequence $\{a_n\}$ starts from $a_{2M+1}=M(2M+1)$ and $a_k$ is the largest multiple of $k$ no more than $a_{k+1}+M$, i.e. $$a_k=k\left\lfloor\frac{a_{k+1}+M}{k}\...
Andrew Sakura's user avatar
75 votes
3 answers
3k views

Is there any deep philosophy or intuition behind the similarity between $\pi/4$ and $e^{-\gamma}$?

Here is a couple of examples of the similarity from Wikipedia, in which the expressions differ only in signs. I encountered other analogies as well. $${\begin{aligned}\gamma &=\int _{0}^{1}\int _{...
Anixx's user avatar
  • 9,316
67 votes
2 answers
4k 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 ...
Notamathematician's user avatar
64 votes
5 answers
16k views

The unreasonable effectiveness of Padé approximation

I am trying to get an intuitive feel for why Padé approximation works so well. Given a truncated Taylor/Maclaurin series it "extrapolates" it beyond the radius of convergence. But what I can'...
Felix Goldberg's user avatar
59 votes
8 answers
13k views

Series whose convergence is not known

For most of the mathematical concepts I learn, it has more or less always been possible to find (at least google and find) unsolved problems pertaining to that specific concept. Keeping a bag of ...
Chulumba's user avatar
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55 votes
1 answer
3k views

A mysterious connection between primes and $\pi$

The Prime Number Theorem relates primes to the important constant $e$. Here I report my following surprising discovery which relates primes to $\pi$. Conjecture (December 15, 2019). Let $s(n)$ be ...
Zhi-Wei Sun's user avatar
  • 14.4k
54 votes
8 answers
9k views

Does the formal power series solution to $f(f(x))= \sin( x) $ converge?

I have spent some time using gp-pari. There is, of course, a formal power series solution to $ f(f(x)) = \sin x.$ It is displayed below, identified by the symbol $g$ because I am not entirely sure ...
Will Jagy's user avatar
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53 votes
3 answers
4k views

Does every real function have this weak continuity property?

In my research I came across the following question : Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}$, there exists a real sequence $(x_n)_n$, taking infinitely many values, ...
Dattier's user avatar
  • 3,737
53 votes
1 answer
3k views

On the first sequence without triple in arithmetic progression

In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence: It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
Sebastien Palcoux's user avatar
51 votes
4 answers
5k 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 $\...
Tito Piezas III's user avatar
50 votes
1 answer
2k 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) = \sum_{n=0}^...
echinodermata's user avatar
48 votes
6 answers
6k 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 "...
45 votes
5 answers
3k views

How many rearrangements must fail to alter the value of a sum before you conclude that none do?

This will not be altogether unrelated to this earlier question. For which classes $C$ of bijections from $\{1,2,3,\ldots\}$ to itself is it the case that for all sequences $\{a_i\}_{i=1}^\infty$ of ...
Michael Hardy's user avatar
44 votes
5 answers
3k views

Fibonacci series captures Euler $e=2.718\dots$

The Fibonacci recurrence $F_n=F_{n-1}+F_{n-2}$ allows values for all indices $n\in\mathbb{Z}$. There is an almost endless list of properties of these numbers in all sorts of ways. The below question ...
T. Amdeberhan's user avatar
44 votes
2 answers
6k views

Is $\sum_{k=1}^{n} \sin(k^2)$ bounded by a constant $M$?

I know $\sum_{k=1}^{n} \sin(k)$ is bounded by a constant. How about $\sum_{k=1}^{n} \sin(k^2)$?
npbool's user avatar
  • 553
43 votes
2 answers
6k views

Alternating sum of square roots of binomial coefficients

Let $$ c_n = \sum_{r=0}^n (-1)^r \sqrt{\binom{n}{r}}. $$ It is clear that $c_n = 0$ if $n$ is odd. Remarkably, it appears that despite the huge positive and negative contributions in the sum ...
Mark Wildon's user avatar
  • 10.8k
43 votes
3 answers
2k views

Proving $\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\frac{x_{i}}{x_{j}}\right\}\le \frac{9}{14}n^2$?

For any postive integer $n$ and for any postive real numbers $x_{1},x_{2},\cdots,x_{n}$, show that $$\sum_{i=1}^{n}\sum_{j=1}^{n}\left\{\dfrac{x_{i}}{x_{j}}\right\}\le \dfrac{9}{14}n^2$$ Let \begin{...
math110's user avatar
  • 4,220
42 votes
2 answers
2k views

Numbers that are generic w.r.t. exponentiation

This is a follow-up to my old question Number of distinct values taken by $x\hat{\phantom{\hat{}}}x\hat{\phantom{\hat{}}}\dots\hat{\phantom{\hat{}}}x$ with parentheses inserted in all possible ways. ...
Vladimir Reshetnikov's user avatar
40 votes
3 answers
1k views

Characterizing positivity of formal group laws

The formal group law associated with a generating function $f(x) = x + \sum_{n=2}^\infty a_n \frac{x^n}{n!}$ is $$f(f^{-1}(x) + f^{-1}(y)).$$ In my thesis, I found a large number of examples of ...
Jair Taylor's user avatar
38 votes
13 answers
7k views

What Are Some Naturally-Occurring High-Degree Polynomials?

To construct J. H. Conway's look-and-say sequence, begin by putting down a 1 as the first entry. The other entries are found by saying the previous entry aloud, and writing what you hear. ...
38 votes
5 answers
6k views

Probabilities in a riddle involving axiom of choice

The question is about a modification of the following riddle (you can think about it before reading the answer if you like riddles, but that's not the point of my question): The Riddle: We assume ...
Denis's user avatar
  • 1,291
36 votes
2 answers
3k views

Why does this sequence converges to $\pi$?

One of my daughters was having a small programming exercise. Let's consider following algorithm: Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$. Remove every $2$nd number. From the ...
mathcounterexamples.net's user avatar
35 votes
7 answers
4k 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 $(p,p+4,p+...
34 votes
1 answer
3k views

A remarkable almost-identity

OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$. Mikhail Kurkov noticed that it ...
Robert Israel's user avatar
34 votes
2 answers
1k views

Representations of $\zeta(3)$ as continued fractions involving cubic polynomials

$\zeta(3)$ has at least two well-known representations of the form $$\zeta(3)=\cfrac{k}{p(1) - \cfrac{1^6}{p(2)- \cfrac{2^6}{ p(3)- \cfrac{3^6}{p(4)-\ddots } }}},$$ where $k\in\mathbb Q$ and $p$ is a ...
Wolfgang's user avatar
  • 13.2k
34 votes
1 answer
2k views

How to prove the identity $L(2,(\frac{\cdot}3))=\frac2{15}\sum\limits_{k=1}^\infty\frac{48^k}{k(2k-1)\binom{4k}{2k}\binom{2k}k}$?

For the Dirichlet character $\chi(a)=(\frac a3)$ (which is the Legendre symbol), we have $$L(2,\chi)=\sum_{n=1}^\infty\frac{(\frac n3)}{n^2}=0.781302412896486296867187429624\ldots.$$ Note that this ...
Zhi-Wei Sun's user avatar
  • 14.4k
32 votes
3 answers
2k views

Are these fast convergent series for $\log(2)$, $\log(3)$ and $\log(5)$ already known and proven?

Updated on Feb.16.2024 Fortunately for three of these series, Eqs. (1), (3) and (4), I have found a proof using classical methods which I placed in the Answers section below. (I doubt that there is a ...
Jorge Zuniga's user avatar
  • 2,202
32 votes
0 answers
2k views

The easily bored sequence

If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
Alessandro Della Corte's user avatar
31 votes
5 answers
1k views

On average, how many uniformly random real numbers $u$ are needed for their sum to exceed $1$, if $u_1$ is in $(0,1)$ and $u_k$ is in $(0,eu_{k-1})$?

A well-known question is: on average, how many uniformly random real numbers in $(0,1)$ are needed for their sum to exceed $1$? The answer is $e$. Let's tweak this question by making each random ...
Dan's user avatar
  • 2,341
31 votes
0 answers
2k views

A question related to the Hofstadter–Conway \$10000 sequence

The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
Alkan's user avatar
  • 681
30 votes
1 answer
1k views

Rearrangements that never change the value of a sum

I posted this question on math.stackexchange.com and so far the only answer posted (also mentioned in the comments under the question) shows that one of my rash initial guesses about the bottom-line ...
Michael Hardy's user avatar
29 votes
2 answers
16k views

power series of the reciprocal... does a recursive formula exist for the coefficients [closed]

Let $f(x)=\sum _{n=0}^{\infty } b_nx^n$ and $\frac{1}{f(x)}=\sum _{n=0}^{\infty } d_nx^n$. Then the coefficients of the reciprocal of $f(x)$ can be written down. The first few terms are: $d_0 = \frac{...
AUK1939's user avatar
  • 569
29 votes
3 answers
3k views

An explicit series representation for the analytic tetration with complex height

Tetration is the next hyperoperation after more familiar addition, multiplication and exponentiation. It can be seen as a repeated exponentiation, similar to how exponentiation can be seen as a ...
Vladimir Reshetnikov's user avatar
28 votes
4 answers
3k views

The function $\sum_{0}^{\infty} x^n/n^n$

The function $F(x) = \sum_{0}^{\infty} x^n/n^n$ may be familiar to many readers as an example sometimes used when teaching tests for absolute convergence of entire functions defined by power series. I ...
Gene Ward Smith's user avatar
28 votes
4 answers
2k views

Advanced software for OEIS?

Is there (if not, why?) a software where I can input a sequence of integers, like into the OEIS, and then it makes some simple transformations on it to check whether the sequence can be obtained from ...
domotorp's user avatar
  • 18.3k
28 votes
2 answers
2k views

"Are you taller than the average of those who are taller than the average?"

I've met tall people. That is: people taller than the average. Every now and then we encounter really tall people, even taller than the average of tall people i.e. taller than the average of those who ...
Qfwfq's user avatar
  • 22.7k
28 votes
0 answers
691 views

Does this infinite primes snake-product converge?

This re-asks a question I posed on MSE: Q. Does this infinite product converge? $$ \frac{2}{3}\cdot\frac{7}{5}\cdot\frac{11}{13}\cdot\frac{19}{17}\cdot\frac{23}{29}\cdot\frac{37}{31} \cdot \cdots \...
Joseph O'Rourke's user avatar
27 votes
3 answers
936 views

A point set of power series with coefficients in {-1, 1}. Connected or not?

Let $z$ be a fixed complex number with $|z|<1$ and consider the set $$X_z := \Big\{\sum\limits_{i=1}^{\infty} a_i z^i \ \Big|\ a_i\in \{-1,1\} \forall i\Big\}.$$ What can be said about the set $M$ ...
Kirby Lee's user avatar
  • 373
27 votes
5 answers
3k views

How to show a function converges to 1

Consider the following recurrence relation in two variables: $$f(a, b) = \frac{a}{a+b} f(a-1,b)+ \frac{b}{a+b}f(a+1,b-1) $$ for positive integers $a$ and $b$, with the boundary conditions $f(0,b)=0$ ...
Simd's user avatar
  • 3,195
26 votes
4 answers
2k views

For $x$ irrational, is $a_{n} =\sum_{k=1}^{n}(-1)^{⌊kx⌋}$ unbounded?

For $x$ irrational, define $a_{n} :=\sum_{k=1}^{n}(-1)^{⌊kx⌋}$. Can you prove that $\left\{a_n\right\}$ is unbounded? I feel that it is not easy to treat every irrational $x$. I have asked in S.E. ...
Chennes's user avatar
  • 385
26 votes
1 answer
7k views

Elegant recursion for A301897

Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here $$a(n)=\frac{1}{n+1}\binom{2n}{...
Notamathematician's user avatar
26 votes
1 answer
1k 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 = \frac{123826979}{...
Paramanand Singh's user avatar
25 votes
4 answers
3k views

Can a conditionally convergent series of vectors be rearranged to give any limit?

Warmup (you've probably seen this before) Suppose $\sum_{n\ge 1} a_n$ is a conditionally convergent series of real numbers, then by rearranging the terms, you can make "the same series" converge to ...
Anton Geraschenko's user avatar
25 votes
2 answers
3k views

Is there any sequence $a_n$ of nonnegative numbers for which $\sum_{n \geq 1}a_n^2 <\infty$ and $\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty$?

Is there any sequence $a_n$ of nonnegative numbers for which $\displaystyle\sum_{n \geq 1}a_n^2 <\infty$ and $$\sum_{n \geq 1}\left(\sum_{k \geq 1}\frac{a_{kn}}{k}\right)^2=\infty\quad?$$ See ...
a_MSE_user's user avatar
25 votes
1 answer
717 views

"Harmonacci" recurrence and identities for $\pi$

While playing with something totally irrelevant I stumbled upon the recurrence: $$a_{n+1} = \frac{1}{a_n} + a_{n-1}$$ It turns out that given $a_0 = 1, a_1 = 1$, $$lim \frac{a_{2n}}{a_{2n-1}} = \...
Victor P's user avatar
  • 353
25 votes
3 answers
2k views

Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim_{n\to\infty}\prod_{k=1}^n f(k/n)<\infty$?

Does there exist a continuous function $f(x)$ such that $f(0)=0$ and $0<\lim\limits_{n\to\infty}\prod\limits_{k=1}^n f(\frac{k}{n})<\infty$ ? I do not see any reason why such a function could ...
Dan's user avatar
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