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.
1,668
questions
0
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0
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22
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Is $\sum\limits_{k=1}^nk^i=S_3(n)\times\frac{P_{i-3}(n)}{N_i}$ for odd $i>1,\sum\limits_{k=1}^nk^i=S_2(n)\times\frac{P_{i-2}'(n)}{N_i}$ for even $i$?
I asked this question here
When I was in high school, I was fascinated by $\displaystyle\sum\limits_{k=1}^n k= \frac{n(n+1)}{2}$ so I tried to find the general solution for $\displaystyle\sum\...
- 103
3
votes
0
answers
46
views
$R$-recursion for the A249833 (similar to A235129)
Let $a(n)$ be A249833 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A(x) = 1 + \int A(x) + (A(x))^2\log A(x)\,dx
$$
The sequence begins with
$$
1, 1, 2, 7, ...
- 3,018
0
votes
1
answer
89
views
The sequence has a stationary accumulation point
Let $f:\mathbb{R}^n\rightarrow \mathbb{R}$ be a smooth (continuously differentiable), convex function with a non-empty set of minimizers and $\{x^k\}$ be a sequence such that
(a) $\{x^k\}$ has an ...
- 21
2
votes
0
answers
84
views
$R$-recursion for the A235129
Let $a(n)$ be A235129 i.e. an integer sequence such that its exponential generating function $A(x)$ satisfies
$$
A'(x) = 1 + A(x)\exp(A(x))
$$
The sequence begins with
$$
1, 1, 3, 12, 64, 424, 3358, ...
- 3,018
1
vote
0
answers
102
views
Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?
Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
- 151
5
votes
1
answer
218
views
Precise asymptotic estimate of a recurrence sequence involving a square root
Consider a recurrence sequence defined like this:
$$ \begin{cases} x_0 = \varepsilon \\
x_{n+1} = x_n + \varepsilon \sqrt{x_n}.
\end{cases}$$
I am interested in estimating the value of $x_{\...
- 565
5
votes
1
answer
339
views
Closed-form for the number of partitions of $n$ avoiding the partition $(4,3,1)$
Let $a(n)$ be A309099 i.e. the number of partitions of $n$ avoiding the partition $(4,3,1)$.
We say a partition $\alpha$ contains $\mu$ provided that one can delete rows and columns from (the Ferrers ...
- 3,018
2
votes
0
answers
57
views
Elementary recursion for the A258173
Let $a(n)$ be A258173 i.e. sum over all Dyck paths of semilength $n$ of products over all peaks $p$ of $y_p$, where $y_p$ is the $y$-coordinate of peak $p$.
A Dyck path of semilength $n$ is a $(x,y)$-...
- 3,018
18
votes
2
answers
887
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 $f(0,b)=0$ ...
- 3,095
0
votes
0
answers
53
views
Making $x^n$ in Pascal's Trangle [migrated]
Label the n th diagonal $a^n$.
(For example,$1,3,6,19,15...$ will be labeled $a^3$.)
Let $p$ be a integer,and try to add or subtract lists of $a^n$ to get a series like below:
The first term is $1$, ...
- 149
-3
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0
answers
42
views
convergence of a numerical series using information about an entire series [migrated]
I'm on a problem that seems simple but turns out to be a bit twisted.
Let be $\sum_{n\epsilon N }^{}{u_nz^n}$ a power series with radius of convergence ρ = 1. Which of the following statements are ...
- 1
1
vote
1
answer
78
views
Formulas for partial composed product
Let $A(x) = \prod\limits_i (x-\lambda_i)$ and $B(x) = \prod\limits_j (x-\mu_j)$. Then, their composed product is defined as
$$
(A*B)(x) = \prod\limits_{i,j} (x-\lambda_i \mu_j).
$$
Generally, we can ...
- 1,131
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0
answers
168
views
Mean Cauchy sequences
Let $X$ be a $\sigma$-finite measure space. A sequence of functions $f_n \in L^1 (X)$ is said to be Cauchy in mean if
$$\lim_{K \to \infty} \limsup_{N, M \to \infty} \frac{1}{NM} \sum_{i = K+1}^{K + N}...
- 4,232
3
votes
2
answers
126
views
Abel–Plana formula with fractional offset
The Abel–Plana formula compares the sum $\sum_{n=0}^\infty f(n)$ to the integral $\int_0^\infty f(x)\,dx$,
\begin{equation}
\sum_{n=0}^{\infty}f\left(n\right)-\int_{0}^{\infty}f\left(x\right)dx=\frac{...
- 173k
4
votes
1
answer
347
views
Series convergence if $\sum a_n^2 < \infty$
There are quite a few simple results about convergent/divergent series derived from similar ones. Here is a question in the same spirit that I saw posted on another forum. Unfortunately, I don't have ...
- 677
6
votes
1
answer
354
views
How to show that $\log 2(1/2\log 2\log 4 + 1/3\log 3\log 6 + \dotsb) + 1/2\log 2 - 1/3\log 3 + 1/4\log 4 - \dotsb = 1/\log 2$ [closed]
I've been studying Ramanujan's work and I stumbled upon this question in the book: Collected Papers of Srinivasa Ramanujan. In there I found question number 769 which is about an infinite sum with ...
2
votes
1
answer
86
views
Natural density of thickly syndetic set
A syndetic set $S$
is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural ...
1
vote
1
answer
157
views
Maximization of $\ell^2$-norm
Consider for $r,c>0$ the set
$$X_{r,c}=\{x \in \ell^1(\mathbb{N}) \mid \|x\|_1=r,\, \forall i \in \mathbb{N}: |x_i|<c\}.$$
Then I can show that $\inf_{x \in X_{r,c}} \|x\|_2 = 0.$
But is it ...
- 11
2
votes
1
answer
146
views
Does every real number $r\in [0,1]$ have a rational sequence $q_n\to r$ s.t. $q_n$ has (simplified) denominator $n$? [closed]
This seems pretty trivial but I can't seem to figure it out. I think it's obviously true, given an unconstrained convergent sequence we just have to add some filler elements, but I'm having trouble ...
14
votes
0
answers
340
views
Will a unit disk be completely covered by randomly placed disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ with probability $1$?
On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random ...
- 2,193
2
votes
0
answers
114
views
Series acceleration for $\sum_{k=0}^\infty\left(\frac{H^k}{k!}\right)^\beta$, $\beta\ll 1$
The probability mass of the Conway-Maxwell-Poisson variable $K$ is given by
$$
\mathsf P(K=k)=\frac{1}{Z(H,\beta)}\left(\frac{H^k}{k!}\right)^\beta
$$
where
$$
Z(H,\beta)=\sum_{k=0}^\infty\left(\frac{...
0
votes
1
answer
272
views
Uniqueness of the $J$ invariant
It seems that
The $J$ invariant is the unique modular function of weight zero for $\operatorname{SL}(2,\mathbb{Z})$ which is holomorphic away from a simple pole at the cusp such that
$$J(e^{2\pi i/3})...
- 303
5
votes
1
answer
419
views
How can I evaluate the following sum?
While studying sequences and series, I came across summations of geometric series. I am able to derive the sum of a geometric progression and that of arithmetico–geometric sequence.
But taking a step ...
2
votes
0
answers
130
views
How to prove this weighted sum inequality with non-increasing sequences?
Problem
I have two non-increasing sequences, $X = (x_1, x_3, x_5, \ldots, x_{n-1})$ and $Y = (y_1, y_3, y_5, \ldots, y_{n -1})$, $n$ is an even integer. I want to prove this inequality:
$$
\sum_{i=1}^{...
- 29
6
votes
1
answer
305
views
Convergence of derived series
There are quite a few simple results about convergent/divergent series derived from similar ones. So, here is something that I came across recently:
Let $\sum_{n=1}^\infty a_n$ consist of positive ...
- 677
0
votes
1
answer
225
views
Nature of $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $ [closed]
I'm trying to determine the nature of this series $ \sum_{n \geq 1} \frac{ \cos(n) \sin(n+1) }{n} $, but I'm not getting anywhere. I've tried using the Abel and trigonometric formulas, but I can't ...
user516435
0
votes
0
answers
136
views
Dark side of the self-inverse permutation
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
Let
$$
f(n) = 2^{\ell(n)}
$$
Let $p_1(n)$ be an arbitrary self-inverse permutation of the non-negative integers such that $p_1(n)<2^k$ iff $n&...
- 3,018
3
votes
4
answers
398
views
Asymptotic for Ramanujan's $\tau$-function
The Ramanujan's $\tau$-function is defined by
$$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$
where $|q|\lt 1$.
Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
- 303
7
votes
2
answers
365
views
Let $(a_n)_{n\in N}=(1,2,3,4,6,8,9,12,\cdots)$ list the set$\{2^n3^m\mid m,n\in N\}$. Find $α$ such that $(a_n)\alpha\pmod1$ is not equidistributed
Let $$(a_n)_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ such ...
- 173
2
votes
0
answers
124
views
Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?
I am trying to get an asymptotic formula such as
$$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$
where $L_4(s, n)$ is the first $n$...
- 3,058
1
vote
1
answer
210
views
A vanishing sum and related $p$-adic congruences
Recently I had a curious discovery. Namely, I have made the following conjectures.
Conjecture 1. We have the identity
$$\sum_{k=0}^\infty\frac{(10k-1)\binom{3k}k\binom{6k}{3k}}{(2k+1)512^k}=0.\label{1}...
- 14.3k
0
votes
0
answers
49
views
Stolarsky representation from Zeckendorf representation with some pairs of bits inverted
Let $a(n)$ be A200714 i.e. Stolarsky representation interpreted as binary to decimal integers.
Let $b(n)$ be A003714 i.e. Fibbinary numbers (Zeckendorf representation interpreted as binary to decimal ...
- 3,018
6
votes
0
answers
149
views
Tweak the numerators in the alternating harmonic series so that the partial sums alternate across $0$. What's the pattern in the numerators?
I was thinking about the alternating harmonic series: $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$.
I wondered what would happen if we tweak the numerators so that the partial sums alternate between ...
- 2,193
7
votes
1
answer
230
views
Efficiently computing $\sum_k x^{k^2}$ modulo $p$
Let $p$ be prime. There is a whole host of "large" degree polynomials that can be computed efficiently modulo $p$. I was wondering if:
$$q(x) = \sum_{k=0}^{p-1} x^{k^2}$$
is a polynomial ...
- 581
0
votes
0
answers
77
views
Partitions in A237981
Let $T(n,k)$ be A237981 i.e. array: row $n$ gives the NW partitions of n; see Comments. Here by $T(n,k)$ I mean $k$-th partition in $n$-th row.
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
...
- 3,018
1
vote
1
answer
95
views
How to prove this iterative convergence of trigonometric functions [closed]
Consider the set of sequences $ S=\{\{s_n\}_{n \ge 0}\mid s_n \in \{-1,+1\}\} $
For any set: $ s=\{s_n \}_{n \ge 0}\in S$, we define the sequence
$$ c_n=\sum_{k=0}^n \frac{s_0s_1 \dots s_k}{2^k}$$
Now,...
- 75
0
votes
3
answers
164
views
Evaluating a sinusoidal series
Define the sequence of functions
$$f_n(x)=\sum_{m=n}^\infty(-1)^m\frac{x^{2m}}{(2m+1)!} {m \choose n} $$
Is there a closed form expression for arbitrary $n$? It is clear that the result should assume ...
- 141
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votes
0
answers
301
views
A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$
Recently, I found a (conjectural) new series for $\sqrt3\pi$:
$$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$
The series converges fast ...
- 14.3k
2
votes
1
answer
345
views
Implementing the $\pi$ BBP algorithm
The formula
$$\pi = \sum_{k=0}^\infty \frac{1}{16^k} \left( \frac{4}{8k + 1} - \frac{2}{8k + 4} - \frac{1}{8k + 5} - \frac{1}{8k + 6}\right)$$
is a basis of the BBP algorithm for calculating arbitrary ...
- 303
3
votes
1
answer
125
views
Weaker version of the lemma of K.L. Chung
Let $\{u_n\}_{n\in\mathbb{N}}$ be a sequence of nonnegative real numbers (i.e., $u_n\geq 0$ for all $n\in\mathbb{N}$). Assume furthermore that, for some positive constant $C$, the following holds:
$$...
- 132
2
votes
2
answers
191
views
An identity for the ratio of two partial Bell polynomials
Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that
the ...
- 796
0
votes
0
answers
68
views
Recursions for the A111528
Let $T(n,k)$ be A111528 i.e. square table, read by antidiagonals, where the g.f. for row $n+1$ is generated by
$$
xg_{n+1}(x) = \frac{1}{n+1}\left(1+nx - \frac{1}{g_n(x)}\right), \\
g_0(x) = \sum\...
- 3,018
0
votes
1
answer
96
views
If $a_1=1$ and $a_n=\sec (a_{n-1})$ then what does the proportion of positive terms approach, as $n\to\infty$?
Consider the sequence $a_1=1$ and $a_n=\sec (a_{n-1})$ for $n>1$.
What does the proportion of positive terms approach, as $n\to\infty$?
At first I thought the limiting proportion might be $\frac{...
- 2,193
13
votes
1
answer
360
views
Four new series for $\pi$ and related identities involving harmonic numbers
Recently, I discovered the following four new (conjectural) series for $\pi$:
\begin{align}\sum_{k=1}^\infty\frac{(5k^2-4k+1)8^k\binom{3k}k}{k(3k-1)(3k-2)\binom{2k}k\binom{4k}{2k}}&=\frac{3\pi}2,\...
- 14.3k
6
votes
0
answers
219
views
A curious series for $L(2,(\frac{-3}{\cdot}))$
Let
$$K:=L\left(2,\left(\frac{-3}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac k3)}{k^2}=\sum_{j=0}^\infty\left(\frac1{(3j+1)^2}-\frac1{(3j+2)^2}\right),$$ where $(\frac k3)$ is the Legendre ...
- 14.3k
5
votes
1
answer
391
views
Three conjectural series for $\pi^2$ and related identities
Recently, I found the following three (conjectural) identities for $\pi^2$:
$$\sum_{k=1}^\infty\frac{145k^2-104k+18}{k^3(2k-1)\binom{2k}k\binom{3k}k^2}=\frac{\pi^2}3,\tag{1}$$
$$\sum_{k=1}^\infty\frac{...
- 14.3k
6
votes
0
answers
272
views
A new series for $\sqrt3/\pi$?
Recently, I conjectured the following identity:
$$\sum_{k=0}^\infty\frac{(66k^2+37k+4)\binom{2k}k\binom{3k}k\binom{4k}{2k}}{(2k+1)729^k}=\frac{27\sqrt3}{2\pi}.\tag{1}$$
This can be easily checked ...
- 14.3k
3
votes
0
answers
264
views
Can you prove and/or generalize this formula involving the Möbius function at n = square free numbers for elliptic curve related sequence in the OEIS?
Let $g(n)$ be the Dirichlet inverse of the Euler totient function:
$$g(n) = \sum\limits_{d|n} d \cdot \mu(d)$$
and let $f(x,y)$ be the elliptic equation:
$$f(x,y)=x^3 - x^2 - y^2 - y$$
Show that the ...
- 1,133
3
votes
0
answers
112
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 3,018
2
votes
1
answer
279
views
Can you confirm the positivity of a quantity involving the Stirling numbers of the first kind
Let $s(m,n)$ denote the Stirling numbers of the first kind. For $m,n\in\mathbb{N}$, define
\begin{equation}
\mathcal{Q}(m,n)=(-1)^n\sum_{\ell=0}^{2n} \binom{m+\ell-1}{m-1} s(m+2n-1,m+\ell-1)\biggl(\...
- 796