As of May 31, 2023, we have updated our Code of Conduct.

# 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.

1,537 questions
Filter by
Sorted by
Tagged with
50 views

### convergence for a series

Show that the series $$\sum_{n=2}^{\infty}\frac{1}{[\frac{(1+\epsilon)\log n}{\log\log n}]!}$$converges for $\epsilon>0$. Stirling's approximation gives that the convergence for the series is ...
53 views

### You have $n$ rectangles of area $1$ and variable height. Pack as many as possible in a semicircle of area $n$. How many leftovers will there be?

You have $n$ rectangles of area $1$ and variable height. Pack as many of these rectangles as possible in a semicircle of area $n$. How many leftover rectangles will there be, in terms of $n$? How to ...
146 views

### Using Ramanujan-type "Legendrian" sequences to find new formulas for $\frac1{\pi}$?

Edit: "To my serial downvoter: Four downvotes in a row. I wrote the moderators voicing my suspicion it is from one person, and they have confirmed it is indeed just one person. Since you have not ...
121 views

### On four Ramanujan-type "Legendrian" sequences with a 3-term recurrence?

I. Recurrences In a previous post, it was mentioned how Almkvist-Zudilin did a computer search for solutions to the recurrence relation, $$(n+1)^3s_{n+1}=(2n+1)(an^2+an+b)s_n+c\,n^3s_{n-1}$$ within a ...
1 vote
45 views

149 views

### Are the numbers $\sum_{n=1}^\infty\frac1{p(n)}$ and $\sum_{n=1}^\infty\frac1{q(n)}$ transcendental?

For each positive integer $n$, let $p(n)$ be the number of partitions of $n$ (i.e., the number of ways to write $n$ as a sum of positive integers), and let $q(n)$ be the number of strict partitions of ...
343 views

### On H. Cohen's four continued fractions for $\zeta(3), \zeta(5), \zeta(7)$?

After 6 years from this old MO post, I finally find in the literature polynomials of deg-$5$ for the continued fraction of $\zeta(5)$. I. Recurrences involving $\zeta(5)$ In Cohen's 2022 paper, ...
113 views

### Finding similar Zudilin-Cohen recurrence relations and cfracs for $\frac{\zeta(4)}{13}$?

I. Two recurrence relations The first one was also discussed in this MO post. We have the similar, \begin{align} (n+1)^5 u_{n+1} &= (2n + 1)(9n^2 + 9n + 3)(15n^2 + 15n + 4)u_n +3n^3(9n^2-1)u_{n-1}\...
78 views

### Simplification of summation and reverse search

Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$. Let $b(n)$ be an integer sequence such that $$b(2^m(2n+1))=\sum\limits_{k=0}^{m}f(m-k)b(2^kn), b(0)=1$$ Let $s(n,m)$ be an integer ...
205 views

### On the continued fractions using Cooper's sequences $s_7,\, s_{10},\, s_{18}$ and the Zudilin-Cohen sequence

In a previous MO post, H. Cohen suggested Gorodetsky's 2021 paper which discussed $6+6+3=15$ "sporadic sequences". The first 6 are Zagier's sporadic sequences, the second 6 are by Almkvist-...
521 views

57 views

94 views

### Sequences that sum up to the many sequences in the OEIS

Let $$a(n,m,k)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj\geqslant 0]\binom{n}{j}\binom{n+kj}{j-1}(m-1)^{j-1}$$ Here square brackets denote Iverson brackets. There are many sequences in the OEIS that are ...
1 vote
169 views

63 views

1 vote
130 views

98 views

### proving inequality in Riemann zeta function

Recently I have made some interesting observations on the limit $$\lim_{k\rightarrow \infty}{\sum_{n=1}^{k}{\dfrac{(-1)^{n-1}\biggl( \cos \left(\beta\ln(n)\right)\biggr)}{n^{\alpha}}}}.$$ When this ...
72 views

### Uniqueness of the permutation

Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$. Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with 1, 2, ...
62 views

### Recurrence relation involving the $\max$ function (reference request): $p_{n + 1} = p_{n} + \max\{r_{p_{n} - 1} - 1, r_{p_{n}}\}$

Consider two sequences $(p_{n})_{n\in\mathbb{N}}$ and $(r_{n})_{n\in\mathbb{N}}$ such that they satisfy the following recurrence relation: \begin{align*} p_{n + 1} = p_{n} + \max\{r_{p_{n} - 1} - 1, ...
196 views

### Continuous functions on $[0,1]^\omega$ and a product lower bound

I have a concrete question about continuous functions on $X = [0,1]^\omega$ (with the product topology). The map $f:X\to [0, 1]$ given by $(x_i)\mapsto \prod x_i$ is well-defined and Borel but not ...
1 vote
Let $B_k$ be the Bernoulli numbers and let \begin{equation} T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1. \end{equation} Prove the inequality \begin{equation*} \frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
### Is the sequence $\{n/\{ \sqrt{2}n\} \}$ uniformly distributed in $[0,1)$?
A classical theorem of Kronecker says that the sequence $(\{\alpha_1 n\}, \{\alpha_2 n\},\dots,\{\alpha_d n\})$ ($n \in \mathbb{N}$) is uniformly distributed in $[0,1)^d$ provided that \$1,\alpha_1,\...