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 (limits-colimits) instead.
4,306
questions with no upvoted or accepted answers
40
votes
0answers
1k views
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 ...
21
votes
0answers
469 views
Compute $\lim\limits_{n\to\infty}(x_{n+1}-x_n)$ if $x_n =\sum\limits_{k=1}^{n-1}f(\frac kn)$ and $f$ continuous (but not continuously differentiable)
The following question from Furdui's book (Exercise 1.32. page 6) is an "open problem" :
Let $f: [0,1] \to \mathbb{R}$ be a continuous (and not a continuously differentiable) function and ...
18
votes
0answers
408 views
$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{x_1^{505}+\cdots +x_n^{505}}{x_1^{2020}+\cdots +x_n^{2020}}dx_1\cdots dx_n$
Evaluate this multiple integral inside a limit:
$$\lim_{n\to\infty} \underbrace{\int_{0}^{1}\cdots \int_{0}^{1}}_{n}\frac{ \sum _{k=1}^{n}x_k^{505}}{\sum_{k=1}^{n}x_k^{2020}} \mathrm d x_1\cdots \...
17
votes
0answers
403 views
How to evaluate the limit of multifactorial $\lim_{n\to 0} \sqrt[n]{n!!!!\cdots !}$
It is well known that $\displaystyle \lim_{n\to \infty}\sqrt[n]{n!}=\infty$, however, if we let $n\to 0$ we have different result with beautiful combination of $e$ and $\gamma$, that is
$$\lim_{n\to ...
14
votes
1answer
121 views
Evaluate $\lim\limits_{x\to 0 } \frac{ e^{\frac{\ln(1+ax)}{x}} - e^{\frac{a\ln(1+x)}{x}}}{x}$.
Problem
Evaluate
$$\lim_{x\to 0 } \frac{ e^{\frac{\ln(1+ax)}{x}} - e^{\frac{a\ln(1+x)}{x}}}{x}.$$
Solution
For convenience, denote $u(x)=\dfrac{\ln(1+ax)}{x}$ and $v(x)=\dfrac{a\ln(1+x)}{x}.$ ...
13
votes
1answer
530 views
A difficult contest question from the former Soviet Union
Let $(a_n)$ be a positive sequence such that $\varlimsup\limits_{n\to\infty} a_n^{1/n}=1$ and $\varliminf\limits_{n\to\infty} a_n^{1/n}<1$.
Prove there exists a subsequence $(a_{n_i})$ such that
$...
12
votes
0answers
100 views
Geometric approach to $\lim_{n\to\infty}\left(\frac{x_{n+1}}{x_n}\right)^n$ where $x_1=1$ and $x_{n+1}=\sqrt{1+x^2_n}$?
I was solving a question :
Let $x_1=1$ and $x_{n+1} = \sqrt{1+x^2_n } \ \ \forall \ \ n\in \mathbb{N}$
Then evaluate $$\lim_{n \to \infty} \left( \frac{x_{n+1}}{x_n} \right)^n$$
The way I did it was ...
11
votes
0answers
185 views
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 ...
10
votes
1answer
185 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 :
$$\forall n \in \mathbb{N}, ~ u_n = \prod\limits_{k=0}^{n-1} \left[ \: \sin \left( \dfrac{\pi}{2^...
10
votes
0answers
438 views
Methodologies to Evaluate $\lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\cos(x^3/3)\,dx$
In This Answer, I wrote
"It is straightforward to show that $\displaystyle \lim_{L\to \infty}\int_0^\infty \frac{\sin(Lx)}{x}\,\cos(x^3/3)\,dx=\frac\pi2$."
For completeness, I've included the "...
10
votes
1answer
11k views
Proving convergent sequences are Cauchy sequences
Prove that if $x_n \rightarrow a, n \rightarrow \infty$ then $\{x_n\}$ is a Cauchy sequence.
I believe I have found the proof as follows, wondering if there are any simpler methods or added ...
10
votes
0answers
245 views
Asymptotic behavior of the generalized polygamma function
The generalized polygamma function$^{[1]}$$\!^{[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 ...
10
votes
1answer
344 views
L'Hopital quicky
suppose L'Hopital applies and $$\lim_{x\to\infty}\frac{f(x)}{g(x)} = \lim_{x\to\infty}\frac{f'(x)}{g'(x)}$$ under what conditions is it true then that
$$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} }{...
9
votes
0answers
332 views
Generalized limits
Cross-posted to Mathoverflow.
$\DeclareMathOperator{\Lim}{Lim}$
$\DeclareMathOperator{\dom}{dom}$
$\DeclareMathOperator{\shift}{\sigma}$
$\DeclareMathOperator{\cesaro}{C}$
After reading Terry Tao's ...
9
votes
0answers
54 views
Objects whose limiting behaviour resembles a group
Is there a name for a structure that isn't a group, but that begins to behave like a group the more operations are performed? I'm trying to take the idea of an attractor from dynamical systems and ...
9
votes
1answer
216 views
How find this value of $A$?
Questionļ¼
Let $z\in C$
Find this value $A$,such
$$\lim_{k\to +\infty}\left(k-\dfrac{W_{k^2}(z)}{W_{k}(z)}\right)= A\cdot i$$
where $i^2=-1$,and
$w_{k}(z)$ is Lambert $W$ function:see http://en....
8
votes
0answers
152 views
Prove that $n\leq a_1+a_2+…+a_n \leq n+1$
If $a_1=2,a_{n+1}=\sqrt{a_n+8}-\sqrt{a_n+3}$. Prove that $n\leq a_1+a_2+...+a_n \leq n+1$ for every $n\ge1$ and $\lim a_n=1$.
I have showed that by squaring and inequality techniques:
$a_i<\sqrt{...
8
votes
2answers
161 views
Prove or disprove: $\lim_{n\to\infty}(a_n n\ln n)=0$, where $a_n>0$, $a_{n+2}-a_{n+1}\geq a_{n+1}-a_n$, and $\sum_{k=1}^{n}a_n$ is bounded
Suppose, for all $n \in \mathbb{N^+}$, it holds that
(1) $a_n>0$;
(2) $a_{n+2}-a_{n+1}\geq a_{n+1}-a_n$;
(3) $\sum_{k=1}^{n}a_n$ is bounded.
Prove or disprove
$$\lim\limits_{n \to \infty} (a_n\...
8
votes
0answers
68 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 ...
8
votes
0answers
131 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 ...
8
votes
0answers
787 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 ...
8
votes
0answers
272 views
Limit approximation for $\pi$ in the four fours puzzle?
The four fours puzzle is a recreational math puzzle whose aim is to express whole numbers using four occurrences of the digit 4 and a specified set of operators. A common variety permits the following:...
7
votes
0answers
56 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 $A$ ...
7
votes
3answers
116 views
$\epsilon - N$ proof of $\sqrt{4n^2+n} - 2n \rightarrow \frac{1}{4}$
I have the following proof for $\lim_{n\rightarrow\infty} \sqrt{4n^2+n} - 2n = \frac{1}{4}$ and was wondering if it was correct. Note that $\sqrt{4n^2+n} - 2n = \frac{n}{\sqrt{4n^2+n} + 2n}$.
$$\left|...
7
votes
0answers
165 views
Evaluate $\lim_{t\to1^-}(1-t)\sum_{r=1}^{\infty}\frac{t^r}{1+t^r}$
$\lim_{t\to1^-}(1-t)\sum_{r=1}^{\infty}\frac{t^r}{1+t^r}$
My approach
$\frac{t^r}{1+t^r}=t^r-t^{2r}+t^{3r}-\cdots$
$\implies \sum_{r=1}^{\infty}\frac{t^r}{1+t^r}=\frac{t}{1-t}-\frac{t^2}{1-t^2}+\...
7
votes
0answers
80 views
Solution verification: evaluate $\lim\limits_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$ where $\lambda>0.$
Problem
Evaluate
$$\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)^{\...
7
votes
0answers
395 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 ...
7
votes
0answers
868 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:$$x_i=0$$ $$v_i=0$$ $$a_i=0$$$$j_i = jā $$.
The product will be moved forward ...
7
votes
1answer
316 views
Oscillating integral converges to zero?
An integral like, say,
$$\int_0^1 \cos[ nf(x)]~dx$$
with some function $f$ which is well behaved, and maybe almost everywhere non-zero, should be very small for large $n$ since the positive and ...
7
votes
0answers
276 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 $x$,
$$ \frac{p}{2\pi}\int_{-\infty}^{+\...
7
votes
0answers
342 views
Definition of the Limit of a Function for the Extended Reals
Definition 4.33 of Rudin's Principles of Real Analysis:
Let $f$ be a real function defined on $E \subset R$. We say that $f(t) \rightarrow A$ as $t \rightarrow x$ where $A$ and $x$ are in the ...
7
votes
2answers
315 views
$\lim_{x\to+\infty}\frac{f(x)}{e^x}=l$, then $\lim_{x\to+\infty}\frac{f'(x)}{e^x}=l$?
Let $f(x)$ be a function, the second derivative $f^{\prime\prime}(x)$ exists, and $\lim\limits_{x\to +\infty}\dfrac{f(x)}{e^x}=l$. $c\gt0$ such that for sufficiently large $x$, $|f''(x)|<c|f'(...
7
votes
1answer
226 views
How can I find the limiting value of a time-dependent PDE?
I've managed to reduce a question in probability to the following simple looking PDE:
$$
u_t = -t u_x + \frac{1}{2} u_{xx}, {\rm ~for~} x>0, \, t \in \mathbb{R} \;,
$$
with a limiting initial ...
6
votes
0answers
166 views
Evaluate $\lim_{z\to\infty} \frac{(1 - z^4)^{1/4}}{z}$.
I'm going through my (applied) complex analysis notes from almost a year ago and trying to make sense of something I wrote down. Of course, anybody's own solution is very much welcome.
We used the &...
6
votes
1answer
169 views
Interpretation of Differentials
$$
\newcommand{\qa}{P}
\newcommand{\qb}{Q}
\newcommand{\da}{dP}
\newcommand{\db}{dQ}
\newcommand{\positiverealnumbers}{\mathbb{R}_+}
\newcommand{\realnumbers}{\mathbb{R}}
\newcommand{\naturalnumbers}{\...
6
votes
0answers
93 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 $$f_\...
6
votes
0answers
96 views
A non-zero continuous function such that summing over equally spaced values always gives zero
Quite some time ago I wondered whether or not there exists some non-zero sequence of real numbers $(a_n)_{n \in \mathbb{N}}$ such that
$$
\sum_{n=1}^{\infty} \, a_{n\times m} = 0,
$$
for all $m \in \...
6
votes
0answers
176 views
Is Hilbert's space-filling curve measure preserving?
Say $f_n:[0,1]\to [0,1]^d$ is the $n$-th iteration of a $d$-dimensional Hilbert curve touring its range. Is it true that for any open $S\subset [0,1]^d$, then amount of time $f_n$ spends in $S$ is ...
6
votes
1answer
138 views
Density of $\{\sin(x^n)|n\in\mathbb{N}\}$ for $x>1$
While reading other topics, e,g, Is $n \sin n$ dense on the real line? or Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $[-1,1]$ for every natural number $m$?, the following problem appeared in ...
6
votes
1answer
306 views
Limit at Infinity of Maclaurin Series
Let $f(x) = \sum_{n=1}^\infty a_n x^n$. What is $\lim_{x\rightarrow \infty} f(x)$ in terms of the $a_i$?
That question may be too broad, so here are some restrictions: Assume f(x) is continuous (and ...
6
votes
0answers
132 views
Limit points of a sequence constructed from pi (if pi is normal)
I'm interested in this question apparently posed by John Nash, which I found in the book A Beautiful Mind.
If you make up a bunch of fractions of pi $3.141592\ldots$. If you start
from the ...
6
votes
0answers
173 views
A generalization of the Euler-Mascheroni constant
Let $f:[1,+\infty)\rightarrow \mathbb{R}$ be a differentiable function. We are dealing with the limit of the sequence
$$
f(n)-\sum_{k=1}^nf'(k).
$$
If $f=\log$, then it is convergent to $-\gamma$ (...
6
votes
0answers
127 views
Divisibility sequence resulting in limit with pi
Consider the following sequence of operations :
Start with a natural number $n$ and then round it up to the closest multiple of $n-1$ .Then round up this new number to the closest multiple of $n-2$...
6
votes
0answers
5k views
Differentiating $\arcsin(x)$ using first principle method.
Q. Differentiate $\sin^{-1}(x)$ using first principle (delta) method.
I did this the following way:
$$y=\sin^{-1}(x)$$
$$\therefore\frac{dy}{dx}=\lim_{h\to \:0\:}\left(\frac{\sin^{-1}\left(x+h\...
6
votes
0answers
638 views
Successive ratios of a sequence, is this limit true?
The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise:
$$\displaystyle T = \left(\begin{matrix} 1&0&0&...
6
votes
0answers
655 views
Limit of fractional part
Prove that the limit as n tends to infinity from
$\{n!\sqrt2\}$ does not exist.
where {} denotes fractional part and "!" denotes factorial.
I don't have many ideas. I would try to show that , given ...
6
votes
1answer
160 views
Closed form of $\lim_{n\to\infty}[(\sum_{k=1}^n\frac{1}{n\ln(1+\frac{k^2}{n^2})})-\frac{{n \pi}^2}{6}]$
While working on a question, I wanted to find a limit in closed form as known numbers but I could not find a way to express it.
$$\alpha=\lim_{n\to\infty}[(\sum_{k=1}^n\frac{1}{n\ln(1+\frac{k^2}{n^2})...
5
votes
0answers
67 views
$\lim\limits_{n\rightarrow +\infty}\int_0^{2k\pi}\sqrt{\sin^{2n}(x)+\cos^{2n}(x)}dx$
I think the limit value of the function $\sin^{2n}(x) + \cos^{2n}(x)$ when $n$ tends to infinity is $0$ for all real values of $x$ except when $x$ is an integral multiple of $\pi$, where it comes out ...
5
votes
0answers
92 views
Proof verification: $\lim\limits_{x\to\infty}f(x)=L\Leftrightarrow \lim\limits_{x\to 0^{+}}f\left(\frac{1}{x}\right)=L$
Problem 81(a) of James Stewart's Calculus: Early Transcendentals 8e asks us to prove that $\lim\limits_{x\to\infty}f(x)=\lim\limits_{x\to 0^{+}}f\left(\frac{1}{x}\right)$, provided that these limits ...
5
votes
1answer
88 views
If $(a_n)$ is a sequence such that $a_n=a_{f(n)}+a_{g(n)}$, where $\lim \frac{f(n)}{n}+\lim\frac{g(n)}{n}<1$, can we claim that $\lim\frac{a_n}{n}=0$?
The inspiration for this question came with an attempt to solve this.
Let $(a_n)_{n\in\Bbb{N}}$ be a sequence of real numbers satisfying $a_n = a_{f(n)} + a_{g(n)}~\forall n\in\Bbb{N}$, where $f, g: \...
