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.
23
votes
0answers
726 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 ...
17
votes
0answers
360 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 let
$...
11
votes
0answers
280 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 "...
11
votes
0answers
386 views
Fabius function and equivalent
The Fabius function $F$ can be defined on $[0,1]$ by
$F(0)=0$
$F(1)=1$
on $[0,\frac{1}{2}]$ $F'(x)=2.F(2x)$
on $[\frac{1}{2},1]$ $F'(x)=2.F(2(1-x))$
It's a known example of a not analytic $C^\...
10
votes
0answers
212 views
Asymptotic behavior of the generalized polygamma function
The generalized polygamma function$^{[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$$
where ...
9
votes
0answers
53 views
Objects whose limiting behaviour resembles a group
Is there a name for a structure that isn't a group, but that begins to behave like a group the more operations are performed? I'm trying to take the idea of an attractor from dynamical systems and ...
9
votes
0answers
2k views
How to take limit *along a path*
So in multivariable calculus for a limit of a function to exist, the limits of the function along all possible paths must exist and equal the same value. But how does one calculate the limit along a ...
8
votes
0answers
109 views
Fraction of $1$s in binary representation of $n!$
I plotted a fraction of $1$s in binary representation of $n!$ (i.e. A079584/A072831) for $n$ from $1$ to $10^4$:
It appears it might converge to some limit for $n\to\infty$. Can we (dis-)prove that ...
8
votes
0answers
409 views
Find $ ? = \sqrt[3] {1 + \sqrt[3] {1 + 2 \sqrt[3] {1 + 3 \sqrt[3] \cdots}}} $
I wonder about a closed form for
$ ? = \sqrt[3] {1 + \sqrt[3] {1 + 2 \sqrt[3] {1 + 3 \sqrt[3] {1 + 4 \sqrt[3] {1 + 5 \sqrt[3] \cdots}}}}} $
To be clear
$$? = \sqrt[3]{ 1 + \color{Red}{1}\sqrt[3]{ 1 ...
8
votes
0answers
233 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:...
8
votes
0answers
2k views
List of techniques to evaluate limits?
I'd like to make a complete list of techniques to evaluate a limit.
Definition of the limit
Continuous functions
Algebra of limits
Addition, multiplication, division
Composition
Inverse function
...
8
votes
0answers
275 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
$...
8
votes
0answers
252 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
$$\lim_{x\to\infty}\frac{\frac{f(x)}{g(x)} }{...
7
votes
0answers
128 views
Cleverly showing that $\lim_{x\to 0}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}=\frac{1}{6}$
$$\lim_{x\to 0}\frac{\textstyle x^{\textstyle(\sin x)^{\textstyle x}}-(\textstyle \sin x)^{\textstyle x^{\textstyle \sin x}}}{\textstyle x^3}=\frac{1}{6}$$
The limit is easy to get results, but ...
7
votes
0answers
145 views
Recurrence $a_{n}=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$
I am considering the sequence
$$a_n=a_{\lfloor 2n/3\rfloor}+a_{\lfloor n/3\rfloor}$$
with $a_0=1$, and I would like to calculate the limit
$$\lim_{n\to\infty} \frac{a_n}{n}$$
I have seen this famous ...
7
votes
0answers
62 views
Solution verification: evaluate $\lim\limits_{n \to \infty}\frac{1^{\lambda n}+2^{\lambda n}+\cdots+n^{\lambda n}}{n^{\lambda n}}$ where $\lambda>0.$
Problem
Evaluate
$$\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
145 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
653 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 ...
7
votes
0answers
261 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 $x$,
$$ \frac{p}{2\pi}\int_{-\infty}^{+\...
6
votes
0answers
94 views
Evaluate $\lim_{t\to1^-}(1-t)\sum_{r=1}^{\infty}\frac{t^r}{1+t^r}$
$\lim_{t\to1^-}(1-t)\sum_{r=1}^{\infty}\frac{t^r}{1+t^r}$
My approach
$\frac{t^r}{1+t^r}=t^r-t^{2r}+t^{3r}-\cdots$
$\implies \sum_{r=1}^{\infty}\frac{t^r}{1+t^r}=\frac{t}{1-t}-\frac{t^2}{1-t^2}+\...
6
votes
0answers
72 views
What is the minimum value of $f_\infty=\frac{x}{\sqrt{x-\sqrt[3]{x-\sqrt[4]{x-\cdots}}}}$?
In a similar vein to What is the maximum value of this nested radical?, I'd like to share a similar nested radical, but this time with changing fractional powers.
What is the minimum value of $$f_\...
6
votes
0answers
56 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 ...
6
votes
0answers
105 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
633 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 ...
6
votes
0answers
129 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
131 views
Is this property of continuous maps equivalent to properness?
For the purposes of my question, a continuous map $f : X \to Y$ is proper if it is closed and the preimage of every compact subspace of $Y$ is a compact subspace of $X$.
Say a continuous map $f : X \...
6
votes
0answers
109 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
496 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:
$$\displaystyle T = \left(\begin{matrix} 1&0&0&...
6
votes
0answers
518 views
Limit of fractional part
Prove that the limit as n tends to infinity from
$\{n!\sqrt2\}$ does not exist.
where {} denotes fractional part and "!" denotes factorial.
I don't have many ideas. I would try to show that , given ...
5
votes
0answers
127 views
A closed form for $\int_0^\pi \lvert \sin(m t) \cos(n t) \rvert \, \mathrm{d} t$
Motivated by this nice question I have been trying to compute the function $f: \mathbb{R}^+ \to \left[0,\frac{1}{2}\right]$ defined by
$$f(\alpha) = \lim_{x \to \infty} \frac{1}{x} \int \limits_0^x ...
5
votes
0answers
123 views
Hermite Polynomial Integral Limit
I'm trying to find the following limit:
$$ \underset{n \to \infty}{\textrm{lim}}
\frac{\sqrt[3]{n}}{2^n n! \sqrt{\pi}}\ \int_{\sqrt{2n+1}}^{\infty}\textrm{H}^2_n(x) e^{-x^2}dx $$
Where H is a ...
5
votes
0answers
89 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 ...
5
votes
0answers
135 views
Does this Abel sum related power series diverge to $-\infty$ at $1$?
Define $a_n$ for $n\geq 2$ to be $1$ if the second most significant digit of $n$ in binary is $1$ and $-1$ otherwise. For example for $n=23$, in binary $n=10111_2$ and so $a_n=-1$, becuase the second ...
5
votes
0answers
101 views
A limit related to asymptotic growth of tetration
The tetration is denoted $^n a$, where $a$ is called the base and $n$ is called the height, and is defined for $n\in\mathbb N\cup\{-1,\,0\}$ by the recurrence
$$
{^{-1} a} = 0, \quad {^{n+1} a} = a^{\...
5
votes
0answers
64 views
Generalized Taylor derivatives test
I am seeking for a proof of the generalized derivative test to find inflection points, minima and maxima.
I am seeking for a proof that I read some time ago but can't find anymore. The thesis was that ...
5
votes
0answers
266 views
Definition of the Limit of a Function for the Extended Reals
Definition 4.33 of Rudin's Principles of Real Analysis:
Let $f$ be a real function defined on $E \subset R$. We say that $f(t) \rightarrow A$ as $t \rightarrow x$ where $A$ and $x$ are in the ...
5
votes
0answers
338 views
Find the limit of $\sum \frac{1}{\log^n(n)}$
Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{\log^n(n)}$$ While proving the convergence ...
5
votes
0answers
82 views
Prove that $||\sum_{n=0}^{+\infty}{x_n}||\le\sum_{n=0}^{+\infty}||{x_n}||$ when series $\sum_{n=0}^{+\infty}{x_n}$ are absolutely converge?
I think it should be proved that:
Since
$$||\sum_{n=0}^{N}{x_n}||\le\sum_{n=0}^{N}||{x_n}||$$
so
$$\lim_{N\to+\infty}||\sum_{n=0}^{N}{x_n}||\le\lim_{N\to+\infty}\sum_{n=0}^{N}||{x_n}||$$
so
$$|...
4
votes
0answers
59 views
Proving $\lim_{x\to\infty}\frac{x+3}{x^2-3}=0$ using delta-epsilon
I'm trying to prove $$\lim_{x\to\infty}\frac{x+3}{x^2-3}=0$$
using delta-epsilon.
In the definition of limit
$$|f(x)-L|\lt\epsilon$$
$$|\frac{x+3}{x^2-3}-0|\lt\epsilon$$
$$|\frac{x+3}{x^2-3}|\...
4
votes
0answers
72 views
Is it possible to prove the derivative of sine geometrically without arc length?
There are a great many ways to prove that the derivative of sine is cosine, some of them based on things like the Taylor series definition. I’d like to prove it using only the right-triangle ...
4
votes
0answers
191 views
A rigorous yet intuitive summary of inflection and critical points for beginning calculus?
I haven't done these in awhile. While my analysis covered continuity but not differentiability, I have so far not revisited these in learning geometry or algebra. I am trying to help a calculus ...
4
votes
0answers
145 views
Limit of matrix function
Let $A\in\mathbb{R}^{n\times n}$ be a matrix whose real eigenvalues have negative real part, and $X=X^\top\in\mathbb{R}^{n\times n}$ be a positive semidefinite matrix, i.e., $X\succeq 0$. Consider the ...
4
votes
0answers
93 views
Prove a limit involving the ceiling function
I found a pattern that I want to prove:
$$f(x) = 2^{\lceil \log_2(3^x)\rceil} - 3^x\quad \{x\in\mathbb{Z}^+\} $$
$$ \lim_{x\rightarrow\infty} f(x) = \infty $$
Discussion:
$$ f(x) = 2^{\lceil \...
4
votes
0answers
45 views
Estimate on Limit of Recursive Sequence
How can I estimate (via a lower bound) the limit of the recursive sequence
$$P_{n+1}=P_n-\frac{C(P_n-1)^2}{(2^n+C)(P_n+C2^{-n})}$$
where $0<C<1$ and $1<P_0<2$. Let $P_{\infty}=\lim_{n\to\...
4
votes
0answers
69 views
A convergent sum of a divergent and convergent sequence?
Is the following argument correct?
Sequences $(x_n)$ and $(y_n)$, where $(x_n)$ converges, $(y_n)$
diverges, and $(x_n+y_n)$ converges.
Proof. The request in question is impossible. Assume that ...
4
votes
0answers
58 views
Nonsmooth data in the conservation laws, their approximations and limits
In the book by R. LeVeque: "Numerical methods for conservation laws", Birkhauser, (1992), 2nd edition, in the Subsection 3.1.2 called "Nonsmooth data", the author talks about possibilities for finding ...
4
votes
0answers
170 views
(Poisson limit theorem) Random variable $X_n$ ~ Bin$(n,p_n)$ convergences to $Z$ ~ Poisson($\lambda$)
$(\star)$ : Let $ X_n $ ~ Bin($n,p_n$) ( $ n \in \mathbb{N} $ )
$ n \cdot p_n \rightarrow \lambda $ for $ n \rightarrow \infty $.
Then $ X_n$ convergences to Poisson $Z$ ~ Poisson($\lambda$).
I ...
4
votes
0answers
224 views
Integral with Riemann Zeta Function. Trying to understand behavior
Hey Math Stack Exchange,
I have an integral that I'd like to try and simplify or get bounds on. It involves the Riemann Zeta function which I'm not super familiar with and so I thought I'd look for ...
4
votes
0answers
136 views
If $Z=X\cdot Y$ then $E[X^k]= \lim_{n \to k} \frac{ E[Z^n]}{ E[Y^n]}$ if $E[X^k]<\infty$
I am curious whether the following theorem is true.
Let $Z=X \cdot Y$ where $X$ and $Y$ are two independent positive
random variables. Moreover, suppose that $E[Z^k],E[Y^k]<\infty$ for
all ...
4
votes
0answers
106 views
showing that a multivariate function is not continuous at the origin, although Wolfram Alpha says it is
I am being asked to show that the following function is NOT continuous at the origin.
$f(x,y) =\begin{cases}\frac{x^4y^5}{x^8 + y^{10}} &&(x,y) \not= (0,0) \\0 &&(x,y) = (0,0) \end{...
