Questions tagged [sequences-and-series]
For questions concerning sequences and series. Typical questions concern, but are not limited to: identifying sequences, identifying terms, recurrence relations, $\epsilon-N$ proofs of convergence, convergence tests, finding closed forms for sums. For questions on finite sums, use the (summation) tag instead.
66,250
questions
-1
votes
0
answers
8
views
Closed form of coefficients of a finite field polynomial
I want to find a valid polynomial for a finite field $\mathbb{Z}_p[x]_{f(x)}$ with $d=deg(f(x))$. For this definition to hold, it can be deduced that $p$ must be prime and the polynomial $f(x)$ ...
- 127
0
votes
0
answers
11
views
Series with Hermite polynomials (Mehler Formula)
After a long calculation, I end up with this series:
$$\sum_{n = 0}^{+\infty} \frac{1}{2^n n! (k^2 - 4(2n+1))} H_n(x) H_n(y), $$
where $H_n(x)$ is the physicist's Hermite polynomial and k is a real ...
- 115
0
votes
0
answers
14
views
Fisher information for Poisson distribution
The context for the question is this paper. I am trying to understand how to get from Eq. (5) to Eq. (7).
For simplicity I will only consider 1 dimension, whereas the equations in the paper are ...
6
votes
1
answer
61
views
True or false: $(a_n)$ is equidistributed in $[0,1]\iff$ for each $x\in [0,1],\vert a_n-x\vert<\frac{1}{n}$ for infinitely many $n\in\mathbb{N}.$
I am wondering if either implication is true:
$$(a_n) \text{ is equidistributed in } [0,1] \iff\text{ For each } x\in [0,1],\ \vert a_n-x\vert<\frac{1}{n}\text{ for infinitely many } n\in\mathbb{N}....
- 21.5k
1
vote
1
answer
82
views
Prove that these following limits are 0 using squeeze theorem
Problems:
Prove that the following limits are $0$.
$\displaystyle \lim_{n\to\infty}\frac{n+\cos(n^2-3)}{2n^2+1}$
$\displaystyle \lim_{n\to\infty}\frac{3^n}{n!}$
$\displaystyle \lim_{n\to\infty}\frac{...
- 21
0
votes
2
answers
55
views
If the logarithm of two sequences gets close together, do the sequences get close together?
I should define what I mean by two sequences $(a_n)$ and $(b_n)$ that are close together. What I mean is that for any $\epsilon > 0$, $\exists N$ such that for $n \geq N$ we have $|a_n-b_n| < \...
- 35
0
votes
0
answers
40
views
Proving $\lim_{k \to \infty} \frac{x_0}{a} \cdot \frac{x_0+1}{a+1} \cdot \frac{x_0+2}{a+2} \cdots \frac{x_0+k-1}{a+k-1}=\left(\frac{x_0}{a}\right)^k$
I am trying to understand why for this sequence:
$$\lim_{k \to \infty} \frac{x_0}{a} \cdot \frac{x_0+1}{a+1} \cdot \frac{x_0+2}{a+2} \cdots \frac{x_0+k-1}{a+k-1}=\left(\frac{x_0}{a}\right)^k$$
I ...
- 3,282
4
votes
1
answer
61
views
Is "$x_n>y_n \text{ holds for arbitrarily large } n \Leftrightarrow \limsup\limits_{n\to\infty} \frac{x_n}{y_n} >1$" true?
Let $\{x_n\}_{n=1}^\infty \subseteq (0,\infty)$ and $\{y_n\}_{n=1}^\infty \subseteq (0,\infty)$ be some positive (possibly unbounded) sequences. I am trying to "cleanly" and "tightly&...
-2
votes
0
answers
19
views
Convergence of a sequence if its subsequences of interval 3 are convergent
Given a sequence $a_n$, is it convergent if its subsequences $\{a_{3n}\},\{a_{3n+1}\}$ and $\{a_{3n+2}\}$ are convergent to the same number?
$$
\lim_{n\to\infty} a_{3n} = \lim_{n\to\infty} a_{3n+1} = \...
- 43
-1
votes
1
answer
42
views
Calculator For Mathematical Sequence [closed]
I'm wanting to calculate a mathematical sequence of adding (54.5782*0.99997105^n) (in which n increases by 1 each time) (until n reaches 436,422).
That looks like
...
2
votes
1
answer
77
views
Does $\sum^{\infty}_{n=1} \exp{\left(\frac{-n^\epsilon}{2\log_2n}\right)}$ converge?
I'm working through the proof for the longest run of heads in a Bernoulli process and I'm having some trouble with the infinite series in the title.
Let $n$ be the total number of tosses in the ...
- 63
3
votes
1
answer
80
views
Are the following 2 equations equal to each other?
I'm in highschool learning about integrals and I wanted to find out what
$$
\int{x}^{x}\,dx
$$
equals. I was wondering if the equation I made below is actually equal to it.
$$
\sum_{k=1}^{\infty} \...
- 31
1
vote
1
answer
36
views
generating function of recursion with a hyperbolic function
I am looking to solve a recursion for a certain sequence $\{a_n\}_{n \geq 1}$ through its generating function
$$f(x)=\sum_{n\geq 1} a_n x^n\tag{1}\label{1},$$
which after plugging-in the specific ...
- 149
3
votes
1
answer
171
views
Is it true, for every $n,m\in\Bbb{N}$ and some election of signs, that $\sum_{i \in \{m+1,...,m+n\} }\pm \frac{1}{i}=0$?
Motivated by Sum of reciprocal of primes I have the question of it is true that for every $n,m \in \mathbb{N}$ we can have
$$\sum_{i \in \{m+1,...,m+n\} }\pm \frac{1}{i}=0$$ for some election of the ...
- 121
0
votes
1
answer
70
views
Is there any counter example that any infinite triangle construction converge?
Given an Initial Triangle $A_1B_1 C_1$ construct the triangle $A_{n+1}B_{n+1}C_{n+1} $ from $A_n B_n C_n$, where $A_{n+1}$ is the $r_1$-th triangle center of $A_n B_n C_n$, $B_{n+1}$ is the $r_2$-th ...
- 6,745
1
vote
0
answers
20
views
Nonnegativity of a Summed Gaussian Difference Series for 1D Dirichlet Heat Kernel
Consider:
$$G(z;\alpha) = e^{-\alpha z^2}$$
Take $x,y \in [0,1]$. Is it possible to show that:
$$ \sum_{n \in \mathbb Z} G(x-y+2n; \alpha) - G(x+y + 2n; \alpha) \geq 0$$
for all $\alpha > 0$? If we ...
- 283
0
votes
0
answers
29
views
Reshuffling and Fubini's theorem
Suppose I have an absolutely convergent series
$$
\sum_{k = 0}^\infty a_k x^{E_k}
$$
but the sequence $\{E_k\}$ of positive real numbers is not nice: it will contains accumulation points and ...
- 31
1
vote
1
answer
53
views
Prove that the inner limit of a set sequence is closed
I am reading the book "Set-Valued, Convex, and Nonsmooth Analysis in Dynamics and Control: An Introduction" by Goebel (https://epubs.siam.org/doi/10.1137/1.9781611977981).
I am confused by ...
- 11
3
votes
0
answers
67
views
Do all tetration sequences converge modulo an integer? [duplicate]
I was messing around with tetration sequences, i.e. sequences of the form,
\begin{equation}
x_n = x \uparrow \uparrow n \mod{c}
\end{equation}
where $x$, $c$, and $n$ are integer numbers. I noticed ...
1
vote
1
answer
36
views
For all $n\in\mathbb{N},\ n+1/2$ is in a $3-$term A.P. with two other members of Szekeres's sequence.
Let $A:=$ Szekeres's sequence = $1,2,4,5,10,11,13,14,28,29,31,32,37,38,\ldots$
Due to the greedy construction of this sequence, appending any number not already in the sequence causes it to no longer ...
- 21.5k
1
vote
1
answer
36
views
On the identity $\sum_{n \leq x} \frac{1}{n} \sum_{k \leq \frac{x}{n}} \frac{\mu(k)}{k}=1$
For all $x\geq 2$
$$
\sum_{n \leq x} \frac{1}{n} \sum_{k \leq \frac{x}{n}} \frac{\mu(k)}{k}=1.
$$
This is an exercise in The Development of Prime Number Theory:
From Euclid to Hardy and Littlewood by ...
- 590
4
votes
0
answers
35
views
Is there a Salem-Spencer set whose sum of reciprocals is greater than that of Szekeres's sequence?
Edit: My question is basically the same as This question, which I think remains unanswered, although in the comments,
ReverseFlowControl claims to have found a sequence, but I don't understand their ...
- 21.5k
-1
votes
0
answers
39
views
Find $\lim_{n\to \infty }\frac{\left(2n^{1/n}-1\right)^n}{n^2}$ [duplicate]
Find $\lim_{n\to \infty }\left(2n^{1/n}-1\right)^n/n^2$.
I can use L'Hospital rule and differentiate it two times. But is there a simple way to solve this?
-1
votes
0
answers
19
views
How many rounds needed to transform a subconstant $\delta_0$ to a constant by iteratively doing $\delta_{i+1} = \delta_i + (1-\delta_i)\delta_i^p/2$
Suppose we have a large constant $p > 1$, and a subconstant $\delta_0 = \delta_0(n) = n^{-0.01}$. In each iteration we obtain a new $\delta$ by
$$ \delta_{i+1} = \delta_i + (1-\delta_i)\delta_i^p/2,...
- 99
1
vote
1
answer
51
views
A confusion about evaluating a sequential limit using the definition of the Riemann integral.
So the question is as below:
$$\lim_{n\to +\infty} \left(\frac{1}{n^2+n+1}+\frac{2}{n^2+n+2}+\frac{3}{n^2+n+3}+\cdots +\frac{n}{n^2+n+n}\right)$$
And my strategy to tackle this is to make use of the ...
- 91
7
votes
4
answers
321
views
Finding out if a series converges without solving linear recurrence relation
I am to study the convergence of the $\sum_{k=1}^{\infty}\frac{a_k}{5^k}$, where $a_{k+2} = 7a_{k+1}-12a_k$, $a_1 = 7, a_2 = 25$.
I have found that the series converges and the value of the sum is $\...
- 253
-3
votes
0
answers
37
views
A tricky series [duplicate]
I am pretty sure I have seen this somewhere before. The problem states
$$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}\sum_{k=0}^{2n}\frac{1}{2n+4k+3}=\frac{3\pi}{8}\log \left(\frac{1+\sqrt5}{2} \right)-\...
- 6,680
0
votes
0
answers
81
views
If $\alpha$ is irrational, $\lim_{n \to \infty} \sin(n \alpha \pi)$ DNE
I've read an answer to this on another post here.
According to the green-checked answer there, let $y=x/2$, and we first know that $a_n = \text{ny} \mod{1}$ (i.e., the fractional part of $ny$) is ...
- 33
0
votes
0
answers
32
views
Find upper bound of convergence rate of sequence under recurrence relation
Let $x_0 > 1$, $a, b \in (0, 1],$ and $c > 0,$ and define a sequence $\{x_n\}_n$ as follows: $$
x_{n+1} = x_n - c (x_n^a - 1)(x_n^b - 1).
$$ Assume that $c$ is small enough (depending on $...
- 43
3
votes
2
answers
126
views
Fixed points of $\tan\sqrt{x}$
This question came in my class test in an MCQ format.
$\DeclareMathOperator{\N}{\mathbb N}$ Let $X=\{x\in\mathbb R^+: \tan(\sqrt{x})= x\}$. Consider the sequence $(b_n)_{n\in\N}$ of real numbers ...
- 2,513
0
votes
1
answer
101
views
Let $A=\frac 1{1^2}+\frac 1{2^2}+\frac 1{3^2}+\cdots$ and $B=\frac 1{1^2}+\frac 1{3^2}+\frac 1{5^2}+\cdots$. Find A/B [closed]
I have actually tried solving this question and have got the answer as 2/1. But, the answer is 4/3, can someone please explain why and how?
$A= \frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\cdots\tag1$
$...
- 35
4
votes
2
answers
74
views
Domain of the Maclaurin Expansion for $\ln(1+x)$
The Maclaurin expansion for $\ln(1+x)$ is:
$$\ln(1+x)=x-\frac{x^2}2+\frac{x^3}3-\frac{x^4}4...\infty$$
In my book, it is given that this infinite series representation is valid only when $x\in (-1,1]$....
- 614
0
votes
0
answers
69
views
Sum of scalar map in $\mathbb{R}^3$
Given the following function defined in $\mathbb{R}^3$, with the restriction $0\leq p\leq 1$:
$$f(x,y,z)=p^{| x+y-z| +| x-y+z| +| -x+y+z| +| x+y+z| +| x-y| +| x+y| +| x-z| +| x+z| +| x| +| y-z| +| y+z|...
- 127
1
vote
2
answers
51
views
Proving the series $\sum _{n=1}^\infty \big( (-1)^n (1-x)x^n \big)$ uniformly converges in $[0,1]$
I am trying to prove the following series of functions uniformly converges in $[0,1]$:
$\sum _{n=1}^\infty \big( (-1)^n (1-x)x^n \big)$
I proved that the limit function is $f(x) = \frac{x^2-x}{1+x}$, ...
- 145
-1
votes
0
answers
56
views
How to check whether the following summation converges or diverges? [duplicate]
How to check that the summation $$\sum_{n=1}^{\infty}\frac{n}{2^{n}}$$ converges or diverges?
I really don't know how to check out whether a summation from $n=1$ to $\infty$ converges or diverges. ...
- 17
2
votes
0
answers
95
views
Show $\sum_{k=0}^{\infty}{\frac{1}{2} \choose k}\frac{(-1)^k}{2k+1}=\sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1}$ [closed]
From the binomial theorem, it can be shown that $\pi=4\sum_{k=0}^{\infty}{\frac{1}{2} \choose k}\frac{(-1)^k}{2k+1}$
However, the Leibniz formula is $\pi=4\sum_{k=0}^{\infty}\frac{(-1)^k}{2k+1}$
I ...
- 55
5
votes
1
answer
311
views
+50
Convergence of $\sin\left(\frac{S}{\sin\left(\frac{S}{\sin(\cdots)}\right)}\right)$
I was framing a question about the number of solutions of the following function defined in $\mathbb R^+$ with $y=\frac{\pi}{4x}$:
$$f(x)=\lim_{n\to \infty} \frac{\sin^{-1}x+x^{2n}\tan^{-1}(x-1)}{x(1+...
- 614
1
vote
1
answer
37
views
Prove the sequence $\lim_{ n \to \infty } a_{n}=1$ given $a_{n}=\frac{2n}{2n+\sin n}$ using definition of limit
I'm trying to prove that the sequence $a_{n}$ converges to 1 where $a_{n}$ is:
$$
a_{n}=\frac{2n}{2n+\sin n}
$$
Where:
$$
\lim_{ n \to \infty } a_{n}=1
$$
But I'm having trouble solving for $N$ and ...
- 117
1
vote
1
answer
33
views
Using the Borel-Cantelli-Lemma to show that numbers 0,...,b-1 appear almost surely infinitely often in the b-adic representation of a real number
following problem:
let $U \sim unif([0,1])$ be a uniformly distributed random variable on the unit intervall $[0,1]$.
The $b$-adic representation of $U$ is then given as $\sum_{n=1}^{\infty} D_n \cdot ...
1
vote
1
answer
61
views
What is $\lim_{n\rightarrow\infty}(L_n)^{\frac1n}$ about the LCM of this special double sequence?
As an intermediate step looking for the irrationality of the Catalan's constant, I need the value of this limit $$\rho=\lim_{n\rightarrow\infty}(L_n)^{\frac1n}$$ where $L_n$ is the least common ...
- 435
0
votes
0
answers
44
views
what $a_i$ ensures that $\sum^{\infty}_{i=0}a_i^2 < \infty$?
What requirements are needed on $a_i$, a non-negative sequence to ensure that $\sum^{\infty}_{i=0}a_i^2 < \infty$?
My answer:
If the series converges then it must be that $a_i^2$ approaches $0$, so ...
- 15
0
votes
1
answer
131
views
If $\sqrt{\frac{2}{\pi}}$ is Diophantine?
Is $\sqrt{\frac{2}{\pi}}$ Diophantine?
Which means does $\sqrt{\frac{2}{\pi}}$ have the property: exist $\tau,\gamma>0, $ for $\forall p,q\in\mathbb{Z}$
$$\left|p-q\sqrt{\frac{2}{\pi}}\right|\geq\...
- 127
-2
votes
0
answers
24
views
Find the value of $\sum_{x,y\ge0}\frac {1}{2^{x+y}+|x-y|}$ [closed]
Find the value of $\sum_{x=0}^\infty \sum_{y=0}^\infty \frac {1}{2^x.2^y +|x-y|}$
I approached this problem by expanding for $y$ first but it is not becoming an infinite G.P.
Can you give me a hint ...
4
votes
2
answers
204
views
Approximating the integral of $\arctan(p(x))$
I am interested in methods to estimate the integral
$$\int_0^1\arctan(p(x))\text{d}x$$
where $p(x)$ is a generic polynomial in $x$.
The motive behind this question is the following problem I ...
- 3,728
5
votes
0
answers
111
views
Infinite sum of $(\sqrt{n^2+1}-n)^{2r}$
I came across the following sum in a statistical physics context,
$$
\sum_{n=\frac{1}{2},\frac{3}{2}\cdots}^\infty \left(\sqrt{\left(\frac{n}{\beta}\right)^2+1}-\frac{n}{\beta}\right)^{2r}=\sum_{n=\...
0
votes
0
answers
51
views
Cauchy product for integer sums
For a sequence $(a_i)_{i \in \mathbb{Z}}$ of real numbers we consider the two sided sum: $$\sum_{i=-\infty}^\infty a_i$$
We say that this sum is (absolutely) convergent, if both parts: $$\sum_{i=0}^\...
- 51
0
votes
0
answers
28
views
How to find a sum of reciprocals of powers of two minus 1 [duplicate]
I came across the following series while solving my math problem:
$$
\sum_{x=1}^\infty \frac{1}{2^x - 1}
$$
While it's easy to check the convergence of the series, finding the sum itself seems to be ...
0
votes
0
answers
29
views
Does $\limsup_{𝑛\to\infty} ( 𝑛*𝑎_𝑛 ) = \infty$ imply that the series $\sum_n 𝑎_𝑛$ diverges? [closed]
Consider the series
$$ ( \sum a_n ) $$ where $$( a_n \geq 0 ) $$ for all $$( n )$$
Assume that
$$
\limsup_{n \to \infty} (n a_n) = \infty.
$$
Does this imply that the series $$ ( \sum a_n ) $$ ...
- 11
0
votes
3
answers
70
views
Prove that the series below converges to $\frac{e}{(e-1)^2}$
I'm given the series $\sum_{n=1}^{\infty} ne^{-n}$ and I want to prove that it converges to $\frac{e}{(e-1)^2}$.
This is the little work I've done:
Consider the sequence of the partial sums $s_n=e^{-1}...
- 23
4
votes
1
answer
99
views
Proving $\sum\limits_{k\ge0}\frac1{2k+2}\left[\frac1{2^{2k}(1-2k)}\binom{2k}k\right]^2=\frac{16}{9\pi}$
Whilst looking at Jack's answer to this question, he claims that
$$\sum_{k\geq 0}\frac{1}{2k+2}\left[\frac{1}{2^{2k}(1-2k)}\binom{2k}{k}\right]^2=\frac{16}{9\pi}$$
and, as suggested by OP, this result ...
- 3,886