New answers tagged real-analysis
0
votes
Evaluating $\int_{0}^{\pi/4} \log(\sin(x)) \log(\cos(x)) \log(\cos(2x)) \,dx$
Almost a decade soon, and still no solution (let's put an end to this story - very elegantly)
A solution by Cornel Ioan Valean (in large steps)
To begin with, I'll consider the trivial integral result,...
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0
votes
Continuity iff lower/upper semi continuity (using $\epsilon-\delta$)
There are a few mistakes in your proof of the second direction of implication, firstly we cannot be guaranteed to find a $t>0$ such that
$$ |f(x)-t|<\frac{\varepsilon}{2} $$
Since for example $f(...
- 1,226
0
votes
Precise definition of an "algebraic function"
All your examples 1. - 5. are algebraic functions, because each of them satisfies a defining irreducible polynomial equation: $f(x)^2-x=0$.
In general, an algebraic function is a $k$-valued function (...
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-3
votes
If $\sum a_n$ converges then so does $\sum \left (1-\frac{1}{n}\right) a_n$
Hint: Using Cauchy-Scwartz it follows that $\sum_n \frac{a_n}{n}\le \sqrt{\sum_{n}\frac{1}{n^2}\sum_{n}a_n^2}$. Show that the right-hand side of the inequality converges as $\sum_n a_n$ converges.
- 16.2k
0
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Inequality problem involving sine function
Some hints:
$\int_0^{\pi/2}\sin=1$
$\int_0^\pi=\sum_{j=1}^n\int_{p_{j-1}}^{p_j}$ for any partition $0=p_0<p_1<\cdots<p_{n-1}<p_n=\pi$ of $[0,\pi]$
Try graphing $|\sin|$ and note the ...
- 32.6k
1
vote
$\lim\limits_{n \to \infty} \lim\limits_{m \to \infty} a_{mn}=\lim\limits_{m \to \infty} \lim\limits_{n \to \infty} a_{mn}=0$ implies $a_{nn} \to 0$?
$a_{nm}=e^{-|n-m|}$ is a counterexample
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5
votes
Accepted
$\lim\limits_{n \to \infty} \lim\limits_{m \to \infty} a_{mn}=\lim\limits_{m \to \infty} \lim\limits_{n \to \infty} a_{mn}=0$ implies $a_{nn} \to 0$?
No, this implication is false. Consider $a_{nn}=1$, $a_{nm}=0$ for $m\neq n$.
- 3,857
1
vote
With this approximation of $\pi$, will this value converge?
For
$$J_n=\int_ 0^1 (x-x^2)\,\pi^x\,dx$$ Mathematica gives a nasty regularized hypergeometric function which reduces to
$$J_n=\pi \, \log ^{-n-\frac{1}{2}}(\pi )\, \Gamma (n+1)\, I_{\frac{1}{2} (2
...
- 249k
0
votes
Is Fermat's theorem about local extrema true for smooth manifolds?
Yes! This is exactly the Lagrange Multiplier Criterion for functions restricted to a smooth manifold (the constraint set). Jerry Shurman's book Multivariate Calculus available for free at
https://www....
- 26
0
votes
Limit of composition function
Consider a proof by contradiction:
If the function $f(x)$ doesn't tend to infinity, then it either tends to a finite value, negative infinity, or has no limit when $x\rightarrow+\infty$
The second one ...
- 98
2
votes
Accepted
Do we have $\log\Gamma(2x+1)-2\log\Gamma(x+1)\ge\log(x^2+1)$ for $x\in[0,1]$?
Proof.
We use the integral representation of $\ln\Gamma(y+1)$:
$$\ln\Gamma(y+1) = \int_0^\infty \frac{1}{t}\left(y\mathrm{e}^{-t} + \frac{\mathrm{e}^{-t(y+1)} - \mathrm{e}^{-t}}{1-\mathrm{e}^{-t}}\...
- 33.2k
1
vote
How do I solve $x^{4^x}=4$?
When you properly wrote $$2^{(2x-1)}\log( x)=\log(2)$$ you made the function much more linear than the original one and this is very good for any root finding method.
By inspection, you know the the ...
- 249k
1
vote
Bounded sequence in $H^s$ that has no convergent subsequence.
I would recommend looking at weak convergence (perhaps it is technically more accurate to call what I am about to describe weak-star convergence, but since we are in a Hilbert space the two notions ...
2
votes
Accepted
Proving that if $s_n=\frac1n\sum_{k=1}^n a_n$ diverges to infinity, then $a_n$ also diverges to infinity.
For an arbitrary sequence $(a_n)$ of real numbers and the sequence $(s_n)$ of the corresponding means $s_n=\frac1n\sum_{k=1}^n a_n$ hold the relationships
$$ \tag{$*$}
\liminf_{n \to \infty} a_n \le \...
- 105k
1
vote
For what $x\in\mathbb{R}\setminus\{0\}$ does generalized continued fraction $x+\frac{x}{x+\frac{x}{x+\frac{x}{x+...}}}$ converge?
Condition for Convergence:
Let $k = x+\frac{x}{x+\frac{x}{x+\ldots}}$.
It follows that $k = x + \frac{x}{k}$.
Solve the equation for $k$.
$$k^2-xk-x=0$$
$$k=\frac{x\pm\sqrt{x^2+4x}}{2}$$
$k$ converges ...
- 602
3
votes
Accepted
Are there any sequences whose sequence of averages $s_n=\frac1n\sum_{k=1}^n a_n$ diverges to infinity but $a_n$ is bounded?
By the triangle inequality,
$$
|\sum_{k=1}^n a_k|\le \sum_{k=1}^n|a_k|.
$$
If $|a_k|\le M$ for all $k$, then the inequality above implies $|s_n| \le \frac{1}{n}\cdot nM= M$, so there is no such ...
- 23.5k
0
votes
Accepted
How do you find solutions to the equation $\frac{x}{y^\frac{x}{y}} = 1$, which involves the Lambert W function?
Recall the definition of Lambert W ... $ue^u = v \Longleftrightarrow u=W(v)$. In this case:
\begin{align}
\frac{x}{y^{x/y}} &= 1
\\
x &= y^{x/y}
\\
x &= \exp \left(\frac{x}{y}\ln y\right)
...
- 107k
1
vote
Find $\lim\limits_{x \to \infty} \int_{0}^1 \dfrac{xt^x}{1+t^x}dt$
An alternative approach to Misha's method is that after we substitute $u=t^{x}$ and get $$\int_{0}^{1}\frac{xt^{x}}{1+t^{x}}dt=\int_{0}^{1}\frac{u^{\frac{1}{x}}}{1+u}du$$ It's enough to show that the ...
- 11
6
votes
Accepted
Find $\lim\limits_{x \to \infty} \int_{0}^1 \dfrac{xt^x}{1+t^x}dt$
If we substitute $u = t^x$ (so $t = u^{1/x}$), then $\mathrm du = x t^{x-1}\,\mathrm dt$, so we get $$\int_0^1 \frac{xt^x}{1+t^x}\,\mathrm dt = \int_0^1 \frac{u^{1/x}}{1+u}\,\mathrm du.$$
As an upper ...
- 134k
2
votes
Accepted
Does the sequence $x_n := n^2/\sqrt{n^6+1} + n^2/\sqrt{n^6+2} + . . . + n^2/\sqrt{n^6+n}$ converge? If it does, what value does it converge to?
Fixing $n \in \mathbb N$, we see $$\frac{n^2}{\sqrt{n^6+k}} \le \frac{n^2}{\sqrt {n^6}} = \frac 1n, \,\,\,\,\,\,\,\, \forall k = 1,\ldots, n.$$ Thus $$x_n = \sum^n_{k=1} \frac{n^2}{\sqrt{n^6 + k}} \le ...
- 15.3k
1
vote
Accepted
Problem in understanding proof of Alternating Series Theorem
Since $(s_{2k})_{k\in\Bbb N}$ is an increasing sequence and $s$ is its limit, then each $s_{2k}$ is smaller than or equal to $s$. A similar argument shows that each $s_{2k+1}$ is greater than or equal ...
- 421k
0
votes
What does it mean for $(T_hf)(\xi) = O(h^K)$ for $K\in \mathbb{N}$ (or $K = \infty$) to hold "uniformly" in $h$, $0 < h \leq 1$?
Suppose $f = f(x_1,x_2,\dots;h)$. If we have an expression of the form $f = O(h^\infty)$, which holds uniformly in $x_1,x_2,\dots$, then it means
$$
\text{for all $N\in\mathbb N$,}\ \exists C_N: \...
- 23.5k
0
votes
Series related to $\,_{2p+1}F_{2p}\left(\left\{\frac{a}{2}\right\}_{2p-1},a,\frac12;a+\frac{1}{2},\left\{1+\frac{a}{2}\right\}_{2p-1};1\right)$
We can have
$$
\,_4F_3\left ( \frac14,\frac14,\frac12,\frac12;1,\frac54,\frac54;1 \right )
=\frac{\Gamma\left ( \frac14 \right )^4 }{64\pi}
+\frac{\Gamma\left ( \frac14 \right )^2 }{4\sqrt{2}\pi^{5/...
- 4,277
0
votes
Limit of $[\frac{1}{x}]$ as $x{\rightarrow0}$
You have proved the statement almost 100%. Notice, you have assumed that if some $L$ exists as a limit for the function $[1/x]$, then $∀ \varepsilon > 0$ there exists $δ > 0$ such that $∀|x|<...
1
vote
Limit of $[\frac{1}{x}]$ as $x{\rightarrow0}$
You are almost done. Just pick $n >L+1, n>\frac 1 {\delta}$ and take $x=\frac 1 n$. Then you get $|x|<\delta$ but you don't have $|[\frac{1}{x}] - L| < 1$.
- 28.5k
2
votes
Accepted
Prove $\int_{0}^{1} \frac{k^{\frac34}}{(1-k^2)^\frac38} K(k)\text{d}k=\frac{\pi^2}{12}\sqrt{5+\frac{1}{\sqrt{2} } }$
Months later, I finally find a self-contained proof(which do lead to a general expression). We have for $\Re(s)\in(0,2)$,
$$
\int_{0}^{1} x^{s-1}K(x)\text{d}x
=\frac{\sin\left ( \frac{\pi s}{2} \...
- 4,277
0
votes
Lower bound Sum of Complex Exponentials using derivatives
The problem I see is that $\delta$ is not well defined in terms of $f$ and $x_0$, because $c$ depends on $x$. But you can modify this because you know that $f^{d+1}$ is continuous, so you can ...
- 3,067
0
votes
Computing the Inverse Fourier Transform of $\tanh(x)/x$
Here I follow msm's helpful answer but fixing a few of the constants I believe are incorrect:
Using op's convention for the Fourier Transform, and like msm letting $a=l_s^{-1}\epsilon$: $$\begin{align}...
0
votes
Is the derivative of differentiable function $f:\mathbb{R}\to\mathbb{R}$ measurable on $\mathbb{R}$?
I have read the only answer posted to this question, but I am afraid it is not right:
Of course, if the derivative of $f$ at a point $x$ is greater than a real number $L$, there will be a natural ...
- 43
1
vote
Is there a proof that continuous linear operators are bounded that uses this line of reasoning?
Firstly, unfortunately there is no such result about continuous images of unit spheres being bounded in infinite dimensions. In fact, a result of Bessaga shows that an infinite dimensional Hilbert ...
- 3,857
5
votes
Accepted
T\F: if $\lim_{k\to\infty}\sum_{i=n_k}^{n_{k+1}-1}|\alpha_i|= 0$ then $\sum \alpha_k$ converges.
There exists a sequence $b_k\to 0$ such that the series $\sum b_k$ is divergent and the partial sums of the series form a bounded sequence. For example $$ b_k={(-1)^l\over 2^l},\quad 2^l\le k<2^{l+...
- 25.1k
2
votes
Trouble with delta epsilon limit proof of form $\frac{a}{bx+c}$
Since the question is how to select a $\delta$ and not which delta works, I'll try to answer that, so it's more like a tutorial and won't give you the particular answer explicitly so that you find it ...
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0
votes
Trouble with delta epsilon limit proof of form $\frac{a}{bx+c}$
I had a very similar (almost identical actually) question in an exam only a few days ago, and like you, I got stuck: the expression just seems complicated and messy to manipulate and deal with, and ...
- 1,226
0
votes
Trouble with delta epsilon limit proof of form $\frac{a}{bx+c}$
Since $ lim (x) = 2 => |x-2| < 1 $
$\\$
=> in this restricted domain, $ x $ belongs to $ (1, 3) $
Let's choose $\delta = 2\epsilon$.
$$ 0 < | x - 2| < \delta = 2\epsilon $$
$ $
$ => \...
7
votes
Accepted
How do I solve $x^{4^x}=4$?
$\def\B{\operatorname B}$
$$x^{4^x}=4\iff 4^{4^{-x}}=e^{\ln(4)e^{-\ln(4)x}}=x$$
Now use the Bell polynomial and
How to solve $x^{y^z}=z$:
$$e^{ae^{bz}}=z=1+\frac1b\sum_{n=1}^\infty \frac{(ae^b)^n}{nn!...
- 10.1k
0
votes
Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$
This is long comment. Document equalities related to Hermite polynomials. Feel free to edit.
It seems linear analysis is not sufficient. Multiplication decreases and convolution increases
variance in ...
- 103
1
vote
Let $f\in C^1(\mathbb{R})\cap L^2(\mathbb R)$ s.t. $f'\in L^2(\mathbb{R})$. Then approximate $f,f'$ by $f_n,f_n'$, where $f_n\in C_c(R).$
Making the hints given in the comments more explicit :
cut off $f$ and make a slow transition to $0$
The idea here is to multiply $f$ by a sequence $(u_n)_n$ of smooth compactly supported functions ...
0
votes
Existence of a curve of finite length on the image of a Sobolev embedding
PARTIAL PROGRESS. As you mention in comments (and you should really edit the question to include this important piece of information), by Fuglede's theorem $f(\gamma(t))$ is absolutely continuous for ...
- 31.5k
0
votes
Determining if a function is Lipschitz
DominikS has already answered most of your concerns, but without proofs, so I'll prove that $\arcsin$ is locally Lipschitz on $(-1,1)$ but not Lipschitz on $(-1,1)$. To see that $\arcsin$ is locally ...
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1
vote
A daunting double integral with a simple closed form
$$\begin{align*}
& \int_0^{\pi/2} \int_0^{\pi/2} \cot (x) \csc ^2(y) \log (\cos (y)) \log \left(1-2 \sin (x)+\sin ^2(x) \csc ^2(y)\right) \, dx \, dy \\
&= -\int_0^\infty \int_0^1 \frac{\log\...
- 16.7k
1
vote
Accepted
How to prove sequence has negative values?
By convergence, for every $\varepsilon > 0$, there exists an $N \in \mathbb{N}$ such that
$$\forall n \in \mathbb{N}, \, n > N, \, | a_n - 0.001| < \varepsilon. $$
Now simply take $\...
- 379
1
vote
Accepted
Holder inequality for 3 functions
Indeed, you can recursively apply the Hölder inequality using associativity to get
$$\|f^2g\|_{L^1(\Omega)}=\|f\cdot (fg)\|_{L^1(\Omega)}\leq \|f\|_{L^\infty(\Omega)}\cdot \|fg\|_{L^1(\Omega)}\leq \|f\...
- 15.8k
0
votes
Prove or disprove: the sequence $a_n = \{ \alpha n \}$ (fractional part) converges if and only if $\alpha \in \mathbb{Z}$
We may distinguish $\alpha$ according to it is rational or not.
If $\alpha=h/k$ for some coprime $h$ and $k$, then for all integer $0\le r<k$ there exists some $n_0$ such that $n_0h\equiv r\pmod k$,...
- 5,911
3
votes
Accepted
How to show that a given function doesn't belong to some $L^p$ space?
A general way to check if a nonnegative function $f$ is in $L^p$ is to check its level sets. It's a theorem that a nonnegative function $f\in L^p$ if and only if
$$
\sum_{n\in\mathbb Z}2^{np}|\...
- 23.5k
4
votes
Accepted
A daunting double integral with a simple closed form
$$\int_0^\frac{\pi}{2}\int_0^\frac{\pi}{2}\cot x \csc^2 y\ln(\cos y)\ln(1-2\sin x+\sin^2 x\csc^2y)dxdy$$
$$\overset{\large \sin x\to x \atop \large \cot y \to y}=-\frac12\int_0^\infty \int_0^1 \frac{\...
- 25.3k
1
vote
Differential of a contracting map is contracting
I don't really understand what you did in your question, but the property you want to prove is false. Take for example the function $f\colon \mathbb R\to\mathbb R$ given by
$$
f(x)=\int_0^x \frac{\sin ...
- 31.5k
1
vote
How do I solve $x^{4^x}=4$?
One to approach this is to let $y=4^x$. That's a nice exponential curve that you can plot easily enough on any graph plotter.
That leaves you with $x^y=4$
Taking logarithms gives you $y \ln x=\ln 4$ ...
- 9,484
0
votes
Confusing notation of sequences in $k$-dimensional Euclidean spaces.
A sequence in $\mathbb{R}^k$ is not a sequence of sequences, it is a sequence of vectors. Usually, when we speak of sequences, it is implied that they have an infinite number of terms. In your ...
- 19.6k
2
votes
A daunting double integral with a simple closed form
I tried a semi-experimental heuristic approach which led me to the announced simple (and beautiful!) result.
It consists of four steps
Let $d$ be the double integral in question.
Step 1: letting ...
0
votes
Determining if a function is Lipschitz
Part 1: I agree with your assessment, and the function is only Lipschitz on any compact subset of $(-1, 1)$ (referred to as locally Lipschitz). This is because the derivative will be bounded on those ...
- 1,135
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