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Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use the tag “limits-colimits” instead.

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Does the average primeness of natural numbers tend to zero?

Note 1: This questions requires some new definitions, namely "continuous primeness" which I have made. Everyone is welcome to improve the definition without altering the spirit of the question. Click ...
Nilotpal Sinha's user avatar
34 votes
0 answers
584 views

An iterative logarithmic transformation of a power series

Consider the following iterative process. We start with the function having all $1$'s in its Taylor series expansion: $$f_0(x)=\frac1{1-x}=1+x+x^2+x^3+x^4+O\left(x^5\right).\tag1$$ Then, at each step ...
Vladimir Reshetnikov's user avatar
16 votes
0 answers
594 views

The limit of $(\sin(n!)+1)^{1/n}$ as n approaches infinity

Calculate the limit $$ \lim_{n\rightarrow\infty}(\sin(n!)+1)^{1/n} $$ or prove that the limit does not exist. This appeared as a problem in my mathematical analysis test, and the answer was that the ...
Mivik's user avatar
  • 429
16 votes
1 answer
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Prove that $F_{2n}(x)=0$ has exactly one root in the interval $x\in(0,1),$ and this root $\to 0$ when $n \to \infty.$

Define $$f_1(x)=x\\f_2(x)=x^x\\\vdots\\f_{n+1}(x)=x^{f_n(x)}$$ Let $F_n(x)=f_n^{'}(x).$ Hence $$F_1(x)=1\\F_2(x)=x^x(1+\log(x))\\\vdots$$ Prove that $F_{2n}(x)=0$ has exactly one root in the ...
lsr314's user avatar
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14 votes
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906 views

Asymptotics of sequence of rational numbers

There is a simple sequence of rational numbers. It starts from $a_1=1$, and then $$ a_{n}=\begin{cases} a_{n-1} &\text{for even }n \\ a_{n-1}-\frac1n a_{\frac{n-1}2} &\text{for odd }...
AAK's user avatar
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14 votes
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Does there exist any $p >0$ such that $\frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty$?

Does there exist any $p >0$ such that \begin{equation*} \frac{1}{n^p \sin(n)} \to 0 \;,n\to+\infty \;? \end{equation*} If there is one, what's the infimum of those $p$? Is it also a minimum? I ...
Bob's user avatar
  • 5,755
12 votes
0 answers
387 views

Surprising approximation of exponential series?

Consider the following expression $$ y_j= \sum_{k=0}^{L} \frac{e^{-\sum_{i=-k}^k(k-|i|)x_{j+i}}-e^{-\sum_{i=-k}^k(k+1-|i|)x_{j+i}}}{\sum_{i=-k}^k x_{j+i}}\tag{1} $$ for $1\leq j \leq L$. Given smooth ...
sam wolfe's user avatar
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12 votes
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317 views

Asymptotic behavior of the generalized polygamma function

The generalized polygamma function$^{[1]}$$\!^{[2]}$ is defined as $$\psi^{(\nu)}(z)=e^{-\gamma\!\;\nu}\;\partial_\nu\!\left(\frac{e^{\gamma\!\;\nu}\;\zeta(\nu+1,z)}{\Gamma(-\nu)}\right),\tag1$$ where ...
Vladimir Reshetnikov's user avatar
11 votes
1 answer
296 views

"Continuous composition" of Lie Bracket

Let $A,X\in M_n(\mathbb{R})$. We denote $[A,X]=AX-XA$ the commutator. It is indeed a Lie Bracket for the matrix Lie algebra. Taking $A$ constant, I'm looking for "the flow" of the commutator....
gdcvdqpl's user avatar
  • 376
11 votes
1 answer
287 views

Simplifying $\prod\limits_{k=0}^{n-1}\left(\sin\frac\pi{2^{k+3}}+\frac1{\sqrt{2}}\right)$

I have recently stumbled upon the sequence $\left( u_n \right)_{n \in \mathbb{N}}$ defined as follows : $$\forall n \in \mathbb{N}, ~ u_n = \prod\limits_{k=0}^{n-1} \left[ \: \sin \left( \dfrac{\pi}{2^...
bghost's user avatar
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11 votes
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A non-zero continuous function such that summing over equally spaced values always gives zero

A long time ago now, I wondered whether or not there exists some sequence of real numbers $(a_n)_{n \in \mathbb{N}}$, different from the zero sequence, such that for any $m \in \mathbb{N}$, $$ \sum_{n=...
Mike Daas's user avatar
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10 votes
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695 views

Generalized limits

Cross-posted to Mathoverflow. $\DeclareMathOperator{\Lim}{Lim}$ $\DeclareMathOperator{\dom}{dom}$ $\DeclareMathOperator{\shift}{\sigma}$ $\DeclareMathOperator{\cesaro}{C}$ After reading Terry Tao's ...
user76284's user avatar
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10 votes
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Methodologies to Evaluate $\lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\cos(x^3/3)\,dx$

In This Answer, I wrote "It is straightforward to show that $\displaystyle \lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\,\cos(x^3/3)\,dx=\frac\pi2$." For completeness, I've included the "...
Mark Viola's user avatar
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10 votes
0 answers
67 views

Objects whose limiting behaviour resembles a group

Is there a name for a structure that isn't a group, but that begins to behave like a group the more operations are performed? I'm trying to take the idea of an attractor from dynamical systems and ...
hasnohat's user avatar
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9 votes
0 answers
204 views

When does the infinite recursive fraction integral $\int_a^b \frac{g(x_1)dx_1}{\int_a^{x_1}\frac{g(x_2)dx_2}{\int_a^{x_2}\cdots}}$ converge?

I write problems for integration bees, and I am considering problems of the form in the title, i.e. $$\int_a^b \frac{g(x_1)dx_1}{\int_a^{x_1}\frac{g(x_2)dx_2}{\int_a^{x_2}\frac{g(x_3)dx_3}{\int_a^{x_3}...
Ninad Munshi's user avatar
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9 votes
0 answers
151 views

Fraction of $1$s in binary representation of $n!$

I plotted a fraction of $1$s in binary representation of $n!$ (i.e. A079584/A072831) for $n$ from $1$ to $10^4$: It appears it might converge to some limit for $n\to\infty$. Can we (dis-)prove that ...
Vladimir Reshetnikov's user avatar
9 votes
0 answers
307 views

Is Hilbert's space-filling curve measure preserving?

Say $f_n:[0,1]\to [0,1]^d$ is the $n$-th iteration of a $d$-dimensional Hilbert curve touring its range. Is it true that for any open $S\subset [0,1]^d$, then amount of time $f_n$ spends in $S$ is ...
Christian Chapman's user avatar
9 votes
0 answers
543 views

Definition of the Limit of a Function for the Extended Reals

Definition 4.33 of Rudin's Principles of Real Analysis: Let $f$ be a real function defined on $E \subset R$. We say that $f(t) \rightarrow A$ as $t \rightarrow x$ where $A$ and $x$ are in the ...
aashtonnz's user avatar
  • 111
9 votes
1 answer
239 views

How find this value of $A$?

Question: Let $z\in C$ Find this value $A$,such $$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$ where $i^2=-1$,and $w_{k}(z)$ is Lambert $W$ function:see http://en....
user avatar
8 votes
0 answers
112 views

When can a double limit $\lim_{k\to\infty}\lim_{m\to\infty}f_{k,m}(x)$ be combined into $\lim_{k\to\infty}f_{k,k}(x)$?

Let $f_{k,m}:\mathbb{R}^n\to \mathbb{R}, \forall k, m \in \mathbb{N}$ and suppose that $\lim_{m\to\infty}f_{k,m} = f_{k,-}$ uniformly in $m$ and $\lim_{k\to\infty}f_{k,m} = f_{-,m}$ at least pointwise....
Epsilon Away's user avatar
  • 1,000
8 votes
0 answers
210 views

Distribution of areas in regular $n$-gon with diagonals, as $n\to\infty$

Consider a regular $n$-gon with all diagonals drawn. Here is an example with $n=10$. What is the distribution of the areas of the regions, as $n\to\infty$ ? That is, if we write down each region's ...
Dan's user avatar
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8 votes
0 answers
206 views

Average length of consecutive numbers which have an increasing number of divisors

Consider the nine consecutive natural numbers starting from $1584614377$. ...
Nilotpal Sinha's user avatar
8 votes
0 answers
79 views

A Family of Limits Leading to an Interesting Function

A while back I got very interested in limits of the form $$ \lim_{n\to\infty} (2A)^n \left (A-\underbrace{\sqrt{a+\sqrt{a+\ldots\sqrt{a+z}}}}_{n\textrm{ radicals}} \right )=f_a^{-1}(z) $$ Where $A$ ...
NMK's user avatar
  • 376
8 votes
0 answers
145 views

Solution verification: evaluate $\lim\limits_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$ where $\lambda>0.$

Problem Evaluate $$\lim_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$$ where $\lambda>0.$ Solution Denote $$S_n:=\sum_{k=1}^{n}\left(\frac{k}{n}\right)^{\...
mengdie1982's user avatar
  • 13.9k
8 votes
0 answers
95 views

Assume $x_n>0$,$x_n+\dfrac{4}{x_{n+1}^2}<3.$ Prove $\lim\limits_{n \to \infty}x_n$ exists and evaluate it.

My Solution Notice that $$x_n+\dfrac{4}{x_{n+1}^2}<3=3\sqrt[3]{\frac{x_n}{2}\cdot\frac{x_n}{2}\cdot\frac{4}{x_n^2}}\leq \frac{x_n}{2}+\frac{x_n}{2}+\frac{4}{x_n^2}=x_n+\frac{4}{x_n^2}.$$ This shows ...
mengdie1982's user avatar
  • 13.9k
8 votes
0 answers
2k views

Movement time of object with constant jerk, limited acceleration and velocity

A product is initially at rest on a conveyor belt: The initial conditions of the product can be described as follows:$$x_i=0$$ $$v_i=0$$ $$a_i=0$$$$j_i = j⋆ $$. The product will be moved forward ...
Luminaire's user avatar
  • 181
8 votes
0 answers
1k views

Pythagoras tree bounding size

The Pythagoras tree is a fractal generated by squares. For each square, two new smaller squares are constructed and connected by their corners to the original square. The angle of the triangle formed ...
qwr's user avatar
  • 10.8k
8 votes
1 answer
446 views

Show that $\lim_{x \to a}{P(x)} = P(a)$ for any polynomial $P(x)$

In my calculus textbook I was given the following Problem: If $P(x)$ is a polynomial, show that $\lim_{x \to a}{P(x)} = P(a)$. I found the following solution here, were a proof was given using ...
Incompl33t's user avatar
7 votes
0 answers
132 views

On the limit of a sum equalling $\pi$

I've been looking at the following sum: $$S=\lim_{n\to\infty}\left(\frac{8}{n^2}\sum_{k=1}^n k\sqrt{\frac{n}{k}-1}\right)$$ Which I have proved converges to $\pi$. My proof remains relatively simple. ...
Jacques Tarr's user avatar
7 votes
0 answers
104 views

Sum of all integers up to $x$ with digit sum $t$

If $S(x,t)$ is the sum of all integers up to $x$ whose sum of digits is $t$, is there a way to calculate it? I mean for high arbitrary $x$. For example, $S(120,11) = 29+38+47+56+65+74+83+92+119 = 603$ ...
MC From Scratch's user avatar
7 votes
0 answers
308 views

Evaluate $\lim_{t\to1^-}(1-t)\sum_{r=1}^{\infty}\frac{t^r}{1+t^r}$

$\lim_{t\to1^-}(1-t)\sum_{r=1}^{\infty}\frac{t^r}{1+t^r}$ My approach $\frac{t^r}{1+t^r}=t^r-t^{2r}+t^{3r}-\cdots$ $\implies \sum_{r=1}^{\infty}\frac{t^r}{1+t^r}=\frac{t}{1-t}-\frac{t^2}{1-t^2}+\...
Makar's user avatar
  • 2,279
7 votes
0 answers
116 views

What is the minimum value of $f_\infty=\frac{x}{\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\cdots}}}}$?

In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers. What is the minimum value of $$f_\...
TheSimpliFire's user avatar
  • 27k
7 votes
0 answers
209 views

Limit points of a sequence constructed from pi (if pi is normal)

I'm interested in this question apparently posed by John Nash, which I found in the book A Beautiful Mind. If you make up a bunch of fractions of pi $3.141592\ldots$. If you start from the ...
prdnr's user avatar
  • 348
7 votes
0 answers
819 views

A sufficient condition for a sequence to converge if arithmetic mean of the sequence converges?

We have a well-known conclusion: If a sequence $\{a_n\}_{n\in\mathbb{N}}$ converges, then the arithmetic mean $\frac{S_n}{n}$ (where $S_n=\sum\limits_{k=1}^na_k$ is the nth partial sum) converges to ...
kellty's user avatar
  • 436
7 votes
0 answers
250 views

A generalization of the Euler-Mascheroni constant

Let $f:[1,+\infty)\rightarrow \mathbb{R}$ be a differentiable function. We are dealing with the limit of the sequence $$ f(n)-\sum_{k=1}^nf'(k). $$ If $f=\log$, then it is convergent to $-\gamma$ (...
M.H.Hooshmand's user avatar
7 votes
0 answers
300 views

Prove: $\frac{p}{2\pi}\int_{-\infty}^{+\infty}\frac{\sin xt}{t\cdot \sin\frac12pt}\sin([\frac xp]+\frac12) pt \, \mathrm dt=\cdots$

Suppose $p>0$, define that $$ g(x)=\begin{cases} p\left\lfloor\frac xp\right\rfloor+\frac p2,x\geqslant0\\\\-g(-x), x<0\end{cases}$$ Prove for all $x$, $$ \frac{p}{2\pi}\int_{-\infty}^{+\...
Faye Tao's user avatar
  • 1,834
7 votes
0 answers
7k views

Differentiating $\arcsin(x)$ using first principle method.

Q. Differentiate $\sin^{-1}(x)$ using first principle (delta) method. I did this the following way: $$y=\sin^{-1}(x)$$ $$\therefore\frac{dy}{dx}=\lim_{h\to \:0\:}\left(\frac{\sin^{-1}\left(x+h\...
Hijaz Aslam's user avatar
7 votes
2 answers
343 views

$\lim_{x\to+\infty}\frac{f(x)}{e^x}=l$, then $\lim_{x\to+\infty}\frac{f'(x)}{e^x}=l$?

Let $f(x)$ be a function, the second derivative $f^{\prime\prime}(x)$ exists, and $\lim\limits_{x\to +\infty}\dfrac{f(x)}{e^x}=l$. $c\gt0$ such that for sufficiently large $x$, $|f''(x)|<c|f'(...
ziang chen's user avatar
  • 7,791
7 votes
1 answer
239 views

How can I find the limiting value of a time-dependent PDE?

I've managed to reduce a question in probability to the following simple looking PDE: $$ u_t = -t u_x + \frac{1}{2} u_{xx}, {\rm ~for~} x>0, \, t \in \mathbb{R} \;, $$ with a limiting initial ...
djws's user avatar
  • 508
6 votes
0 answers
68 views

Questions About Four Definitions of The Upper and Lower Limits of A Sequence

Related questions have been posted here and here. Background I have seen the following four definitions of the upper and lower limits of a sequence from textbooks and MSE posts: Definition 1$\quad$ [...
Beerus's user avatar
  • 1,845
6 votes
0 answers
176 views

Proof of Theorem 1.1 of Analytic Number Theory by Iwaniec & Kowalski

I am not clear about the proof of Theorem 1.1 in the book `Analytic Number Theory' by the authors Iwaniec & Kowalski. They say that if a multiplicative function $f$ satisfies $$\sum_{n\le x}\...
Riemann's user avatar
  • 1,163
6 votes
0 answers
276 views

Show that the limit equals $f'(x)$

Assume that $f$ is continuous and differentiable. Futher, let $g(r,t)\geq0$ be such that for $t>x$ $$ \lim_{r \to \infty} \frac{g(r,t)}{g(r,x)} = 0. $$ Show that $$ \lim_{r \to \infty} \frac{\int_x^...
NPHA's user avatar
  • 339
6 votes
0 answers
141 views

How fast does the coprime probability converge to $6/\pi^2$?

It is known that the probability that two positive integers are coprime is $6/\pi^2$. This is an amazing result. I wanted to see experimentally how the probability converges to $6/\pi^2$, but I found ...
Martin Brandenburg's user avatar
6 votes
0 answers
93 views

Limit of $2^{n^2/2}\sum_{j=1}^{n/2} \sum_{k=1}^{n/2}\left(\cos^2(\frac{j \pi}{n+1}) + \cos^2(\frac{k \pi}{n+1})\right)$ as a double integral

I am currently looking into Dimer coverings and my next step is to find how the following limit is calculated: $$\begin{align*} L &= \lim_{n \to \infty}\frac{1}{n^2}\ln\left(2^{n^2/2}\prod_{j=1}^{...
MarlonButBetter's user avatar
6 votes
0 answers
154 views

Study the convergence of $\sum_{n=1}^{\infty}\frac{\sin (n\sqrt{n})}{\sqrt{n}}.$

I try to use the following result: If $f(x)\in C^1[1,+\infty)$ and $\displaystyle\int_1^{+\infty} |f'(x)|{\rm d}x$ is convergent, then $\displaystyle\sum_{n=1}^{\infty} f(n)$ has the same convergence ...
mengdie1982's user avatar
  • 13.9k
6 votes
0 answers
103 views

A curious limit: $\lim\limits_{n\to\infty}\sum\limits_{i=1}^{n}\left[\left(\frac{n}{n+1-i}\right)\right]^{a}f(i) = c\sum\limits_{i\geq 1}f(i)$

I am trying to prove, for the general case whereby $\zeta(\cdot\,,\cdot)$ is the Hurwitz-Zeta function, and $a\in \mathbb{N}$, that $$\mathcal{L} = \lim\limits_{n\to\infty^{+}}\sum\limits_{i=1}^{n}\...
Brian Constantinescu's user avatar
6 votes
1 answer
229 views

Density of $\{\sin(x^n)|n\in\mathbb{N}\}$ for $x>1$

While reading other topics, e,g, Is $n \sin n$ dense on the real line? or Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $[-1,1]$ for every natural number $m$?, the following problem appeared in ...
tong_nor's user avatar
  • 3,994
6 votes
0 answers
788 views

Weird use of Glasser's Master Theorem

Consider the following enumeration of the rational numbers in $[\,0,1)$: $$0, \frac{1}{2},\frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5}, \frac{...
Jack Tiger Lam's user avatar
6 votes
1 answer
707 views

Limit at Infinity of Maclaurin Series

Let $f(x) = \sum_{n=1}^\infty a_n x^n$. What is $\lim_{x\rightarrow \infty} f(x)$ in terms of the $a_i$? That question may be too broad, so here are some restrictions: Assume f(x) is continuous (and ...
simplet's user avatar
  • 61
6 votes
0 answers
137 views

Divisibility sequence resulting in limit with pi

Consider the following sequence of operations : Start with a natural number $n$ and then round it up to the closest multiple of $n-1$ .Then round up this new number to the closest multiple of $n-2$...
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