Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, and integration through the fundamental theorem of calculus.

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27 views

Is the assumption $y \in C^2$ necessary for the Euler method to be of order $p=1$?

In my Intro to numerical analysis course, we did the following. We stated the initial value problem $\dot{y}=\lambda y+f$, where $f \in C[0,\infty)$, and developed the Euler method. Then proved that ...
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3answers
55 views

How can I plot the set $M:=\left\{\left(\begin{array}{c}x\\y\end{array}\right)\in \mathbb R^2:9x^4-16x^2y^2+9y^4\leq9\right\}$

I need to plot the following set: $$M:=\left\{\left(\begin{array}{c}x\\y\end{array}\right)\in \mathbb R^2:9x^4-16x^2y^2+9y^4\leq9\right\}$$ I have solved the equation $9x^4-16x^2y^2+9y^4-9=0$ for ...
3
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2answers
92 views

Find the series expansion for $\int_0^x e^{-t^2}dt$ for $x\in \mathbb R$.

Find the series expansion for $$\int_0^x e^{-t^2}dt \text{ for } x\in \mathbb R$$ $\textbf{The added question:}$ Is it possible to reverse? If so, if we are given a series expansion what is the ...
2
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0answers
95 views

Application of Weierstrass approximation theorem

How to approximate a continuous function to a desired accuracy using a polynomial? Theorem: For any $\varepsilon > 0$ and $f \in C([a,b])$, there exists a polynomial $p$ such that $\sup_{x \in ...
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1answer
57 views

Maclaurin series of (1+x)^(1/x)

how can i find the Maclaurin series of $f(x)=(1+x)^{1 \over x}$? $f(0)$ is not even defined, or should I define it as $f(0)=e$? I stopped at the first derivative as it gets terribly messy. thank ...
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1answer
327 views

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ always convergent? [closed]

If $\sum a_n$ with $a_n>0$ is convergent, then is $\sum {a_n}^2$ and $\sum \sqrt{a_n}$ and $\sum \sqrt{a_na_{n+1}}$ always convergent? Either prove it or give a counter example. I am thinking ...
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1answer
39 views

$F(x) = \int_0^x \sin{((x+t)^s)} dt$

Let $F(x) = \int_0^x \sin{((x+t)^s)} dt$ , how can i find the derivative with respect to $x$. First i tried to use the fundamental theorem of calculus that asserts that $$\text{if } F(x) = \int_a^x ...
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2answers
79 views

Showing convergence rigourously

So I have $f_n(x) = x^{4n} + \frac1{n^2}$ which I know converges to $f(x)=0$ uniformly on interval $[0,1)$, but how can I show this with rigour? Is this acceptably rigourous? $\lim ...
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4answers
43 views

Lowest possible value of a function with derivative greater than 2

I have the following two problems, and want to attempt to solve them with Mathematical rigour(which I don't yet possess): Suppose that $f$ is differentiable on $[1,4]$ and is such that $f(1) = 10$ ...
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51 views

What is $f$? Finding where a function converges pointwise?

I have a question. Let $f_n(x) = x^{4n} + \frac1{n^2}$. AS $n \to \infty$, $f_n$ converges pointwise to a function $f$ on $[0,1]$ What is $f$? Now if I am understanding correctly, couldn't ...
3
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1answer
92 views

Smooth Manifold, covered by 2 Charts is orientable if the Intersection is Connected

I came across this Question: Atlas on a smooth manifold that contains 2 charts in which Professor Lee commented that this Proposition is true only if the Intersection of the two Maps is connected, so ...
2
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0answers
61 views

How to find out the closed form of a function from its parametric form?

In general suppose that we have a parametric curve given by: $$ x = \phi(t) \\ y = \psi(t) $$ Then if $\phi^{-1}$ exists it is easy to get $y$ as a function of $x$ in closed form: $$ y = ...
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0answers
34 views

Existence of measurable fuction on non-atomic measure space whose integral is infinity

Let $(X,M,\mu)$ be non atomic measure space with $\mu(X)>0.$ Show that there is a measurable function $f:X\to [0,\infty),$ for which $\int f(x)d\mu(x)=\infty.$ No idea at all. I am preparing for ...
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2answers
101 views

Local minimum implies local convexity?

Consider a real function $f$, and suppose it has a local minimum at $a\in \mathbb R$. It typically looks like What hypotheses can be added to $f$ so that there is some $\epsilon >0$ such ...
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3answers
137 views

If $\mu(E) > 0$, $\mu(F) = 0$, is $\mu(E + F) = \mu(E)$?

Let $\mu$ be standard Lebesgue measure. If $E$ is a set of positive measure and $F$ is a set of zero measure, then is it true that $\mu(E + F) = \mu(\{e + f: e \in E, f \in F\}) = \mu(E)$?
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221 views

Real-analytic $f(z)=f(\sqrt z) + f(-\sqrt z)$?

Are there nonconstant real-analytic functions $f(z)$ such that $$ f(z)=f(\sqrt z) + f(-\sqrt z)$$ is satisfied near the real line ? Also can such functions be entire ? And/Or can they be periodic ...
4
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1answer
70 views

Prove g is Lebesgue intergrable

Let $f$ be Lebesgue integrable on $(0, 1)$. For $0 < x < 1$ define g(x) = $\int_x^1t^{-1}f(t)dt$ Prove that $g$ is Lebesgue integrable on $(0, 1)$. $\int^1_0g(x)dx=\int^1_0f(x)dx.$ I am not ...
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237 views

Real-analytic periodic $f(z)$ that has more than 50 % of the derivatives positive?

Im looking for a real-analytic function $f(z)$ such that for any $z$ $1) $$f(z+p) =f(z)$ With $p$ a nonzero real number and where $z$ is close to , or onto the real line such that $z$ is in the ...
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1answer
111 views

Prove Lp is not a metric for 0 < p < 1 [closed]

I think that $\mathbb{L}_p$ fails the triangle inequality $\rho(x,z) \le \rho(x,y) + \rho(y,z)$. How do I prove it? Thanks!
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4answers
53 views

Calculate the original function of a Lipschitz function

Suppose $f(x)$ is $L^1$ such that $$|f(x)-f(y)|\leq |x-y|^2$$ almost everywhere, does it imply $f$ is a constant function almost everywhere?
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1answer
60 views

proof of coarea formula for n dimensional hypersurface in $R^n$

$f:R^n \rightarrow R$ be continuous and summable. please give the proof for these formulas $\int_{R^n}f dx = \int_0^\infty(\int_{\partial B(x_0,r)}fdS)dr$ $\frac{d}{dr}\int_{ ...
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0answers
81 views

Hardy Littlewood maximal function and integral comparison.

Define the Hardy Littlewood maximal function $$g^*(y)=\sup \left\{\frac{1}{|B|}\int_B|g(x)|dx:B\text{ is any open ball containing y}\right\}.$$ For given $x_i,r_i,a_i$, first I have shown that ...
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3answers
59 views

Convergence of the series $\sum\limits_{n=1}^{\infty}nkr^{n-1}$, $|r| < 1$

One series that comes up in actuarial science courses is $$\sum\limits_{n=1}^{\infty}nkr^{n-1},\quad |r| < 1\text{, }k>0\text{.}$$ [Typically $|r| < 1$, but I'm wondering if this is a ...
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0answers
29 views

Modifying a Density Function

Assuming a real an continuous function $f_1(x)$ defined on $\mathbb{R}^+$ which satisfies Probability Density criteria: $$ f_1(x) \geq 0 \quad \forall x \geq 0, \quad ...
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2answers
50 views

A question on differentiable functions

Let $f:\mathbb R \to \mathbb R $ be a function differentiable on [$a,b$] , $f(a)=0$ and there is a real number $A$ such that $|f'(x)| \le A|f(x)| , \forall x\in$ [ $a,b$] , then how do we prove that ...
2
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1answer
72 views

if $f$ is differentiable at $x_0$ then the limit exists

Let $f$ differentiable at $x_0$. Show that the following limit exists $$ \lim_{h\rightarrow0} \frac{f(x_0+h)-f(x_0-h)}{h}$$ If $f$ is differetiable at $x_0$ then it's one-sided derivative exists ...
3
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2answers
126 views

Show that $\lim_{n\rightarrow \infty} \int_0^{\pi/2} 2^n \sqrt{n} \sin^n(x) \cos^{n-2}(x) \; dx = \sqrt{2\pi}$

I wish to show $$\lim_{n\rightarrow \infty} \int_0^{\pi/2} 2^n \sqrt{n} \sin^n(x) \cos^{n-2}(x) \; dx = \sqrt{2\pi}$$ I've tried substitution and integration by parts to get a recursive formula for ...
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0answers
56 views

The relation between the integration on concentric hypercubes

In $R^n$, $Q$ is a hypercube that can be in anywhere in $R^n$. $MQ$ is the concentric hypercube with $Q$, and its radius is M times of the radius of $Q$.How to prove that there exists a constant C ...
2
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2answers
84 views

Proving or disproving an Integral inequality

Is the following inequality true ? $$\sqrt[3]{ \int_{0}^{\pi/4} \frac{x}{\cos^2 x \cos^2 (\tan x) \cos^2 \left(\tan (\tan x)\right) \cos^2 \left(\tan \left(\tan \left(\tan x ...
0
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1answer
60 views

$F(x) = \int_0^{x} t^2 e^{t^2}dt$

Let $y_0 = f''(2) + f'(1) + f(0)$ if $f$ is a real function defined by $f(x) = \int_0^{x} t^2 e^{t^2}dt$. How can I calculate the value of the expression $y_0$. I tried use the fundamental theorem ...
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2answers
47 views

Limit Set of All Real Numbers

Not even sure how to start this one. Does anyone know how to do this? Prove that there exists a sequence such that its limit set is the set of all real numbers Limit set is the set of all ...
4
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1answer
65 views

Operator such as $-1$ is the identity element

Short question Do you know an operator such as $-1$ is the identity element ? Long Question This morning, I had a hard time with identity elements. I'm pretty sure that the following isn't very ...
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2answers
54 views

What are examples of functions that are $L^1$ but not $L^2$ and vice versa?

This exercise is in Stein's Real analysis. Find a function such that $f$ is $L^1(\mathbb{R}^n)$, but not $L^2(\mathbb{R}^n)$ Find a function such that $f$ is $L^2$ but not $L^1$. Hint: ...
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2answers
66 views

Does there exist a Lipschitz map from the unit interval onto the unit square?

It is well-known that continuous space-filling curves exist. But can they be Lipschitz? Specifically, is there a Lipschitz map from [0,1] onto [0,1]x[0,1]?
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1answer
127 views

Can any real number be expressed as an integral combination of $e$ and $\pi$?

Prove or disprove: For any real number $x$, there exist integers $a$ and $b$ such that $ae + b\pi=x$. It certainly seems improbable, but how does one prove it?
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1answer
52 views

Convergence of an Improper Integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$

This is a question from an old exam qualifier: Show that the improper integral $\int_{-\infty}^{\infty}\cos(x\log\left|x\right|)dx$ is convergent. I first notice that \begin{equation*} ...
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1answer
57 views

Q: On proof of the root test

I'm reading Rudin's PMA and am a little bit confused as to his proof of the Root Test $(3.33)$ for the case where $\alpha > 1$. The text says: Let $\alpha = \lim_{n\to \infty}\text{sup } ...
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1answer
42 views

Derivatives and connected sets

I'd like to prove the following two propositions: a) Derivatives maps connected sets into connected sets b) There exist functions which maps connected sets into connected sets that are not the ...
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1answer
32 views

Holder continuity of $\frac{x}{|x|^3} \ast f$ with $f \in C^1_0$ in $\mathbb{R}^3$

Ok, so I need to show that for $f \in C^1_0(\mathbb{R}^3)$ the convolution with $k(x) := \frac{x}{|x|^3}$ is Holder continuous. The exponent doesn't matter much as long as I can bound it using ...
0
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1answer
53 views

Quotient of liminf

Given two sequences $\{a_n\}$ and $\{b_n\}$ with $b_n>0$ for any $n$. Does this hold? $$\lim\inf_{n\rightarrow \infty} \frac{a_n}{b_n}= \frac{\lim\inf_{n\rightarrow \infty} ...
2
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2answers
46 views

Matrices and the Dot Product

Prove that real $n \times n$ matrix $A$ satisfies $Ax \cdot Ay=x \cdot y$ for all $x, y \in \mathbb{R}^n$ if and only if $| Ax| = |x|$. $\textbf{My Attempt}$ Write $Ax \cdot Ay= A^2 ( x \cdot y)$. ...
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1answer
62 views

Prove that $ \left\{\frac{\binom{n+k}{k} }{(n+k)^k}\right\}_{n=1}^{\infty} $ converges to $\frac{1}{k!}$.

Prove that $$ \left\{\frac{\binom{n+k}{k}}{(n+k)^k}\right\}_{n=1}^{\infty} $$ converges to $\frac{1}{k!}$, where $$\binom{n+k}{k} =\frac{(n+k)!}{n!k!}. $$ I'm not even sure how to approach ...
4
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1answer
147 views

Double integral containing $e^{(b+ic)/z^2}$

I want to solve the two integrals \begin{aligned} I_3\,& = \int_{0}^{\infty} ze^{a/z^2 - z^2} dz\\ I_4\,& = \int_{0}^{\infty} \frac{1}{z}e^{a/z^2 - z^2} dz. \end{aligned} where ...
0
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1answer
20 views

Defining $\sin$ using inverse function as teh first step

We compute the length of the piece of the circle between $0$ and $\theta$ for $\theta < \frac{\pi}{2}$ by considering it as the graph of the function $g(y)=\sqrt{1 - y^2}$ as $y$ varies between $0$ ...
1
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1answer
92 views

Prove there's $x_0$ such that $f'(x_0)=0$

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ differentiable at $\mathbb{R}$ and: $$\lim_{x\rightarrow \infty}\left( f(x)-f(-x) \right) = 0$$ Show there's $x_0$ such that $f'(x_0) = 0$. I tried to use ...
0
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1answer
13 views

Analogous for Weierstrass' theorem for functional series.

The Weierstrass' theorem for functional series say the following: Suppose $\{f_n\}$ is a sequence of functions defined on $E$, and suppose $$|f_n(x)|\leq M_n$$ for every $x\in E$ and ...
3
votes
1answer
30 views

Find interval with function that solves ODE $y'(x)=1+(y(x))^2$

Let $g\in C^1(\mathbb{R})$ with $g'\gt 0$ and $g(0)=0$. Show that for the differential equation $$\begin{cases}y'(x) & = \dfrac{1}{g'(y(x))} \\[8pt] y(0) & = 0 \\\end{cases}$$ there exists ...
5
votes
2answers
66 views

line integral: anticlockwise parametrisation in $\mathbb R^3$

Consider $\gamma$ given by the sides of the triangle with vertices $(0,0,1)^t$, $(0,1,0)^t$ and $(1,0,0)^t$. So $\gamma$ runs through the sides of the triangle. Let $f(x,y,z)=(y,xz,x^2)$. I want to ...
3
votes
2answers
41 views

Very easy question regarding the triangle inequality

This question may be far too easy for this site but I always seem to get stuck when it comes to the triangle inequality. For example, I am trying to prove that differentiability implies Lipschitz ...
0
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1answer
50 views

Any Help would thanks : Product of continuous and Riemann integrable function

Let f:[a,b]→R be continuous. Suppose that for every Riemann integrable function g:[a,b]→R the product fg is Riemann integrable and integral a to b fg=0 Prove that f(x)=0 for all x in [a,b] I know ...