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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1answer
51 views

Real Analysis: Show that, if P* is a refinement of P, then ||P ∗ || ≤ ||P||.

Suppose P = {x0, x1, x2, ..., xn} is a partition of the interval [a, b]. Define the mesh of P, denoted ||P|| to be the length of the longest partition interval defined by $$ P : \|P\| = ...
0
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1answer
39 views

Does the closed form of $f(t) = \int \frac{e^{2 \pi i \alpha t}}{e^{2 \pi i \beta t} - 1} dt$ exist?

I have been working on finding close forms of various Fourier series. The general approach is: From the series find the (not necessarily homogeneous) ordinary differential equation for which the ...
6
votes
1answer
52 views

For which $x\in \mathbb{R}$ does $\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$ converge?

I have to study for which values of $x \in \mathbb{R}$ the following series converges: $$\sum_{n=1}^\infty \left(\frac{x^{2n}}{n} - \frac{n^{2x}}{x}\right)$$ I was only able to say that the ...
0
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0answers
32 views

If the derivative tends to infinity near a point, does that mean that the derivative does not exist at that point?

If $f: [0, \infty) \to \mathbb{R}$, $f \in (C^0 [0, \infty)) \cap C^1(0, \infty)) $ and $\lim_{x \to 0^+}f'\to \infty$ does this imply that the derivative (on the right) $f'(0)$ does not exist?
3
votes
1answer
50 views

Geometrical Interpertation of Cauchy's Mean Value Theorem

Cauchy MVT: If functions f and g are both continuous on the closed interval [a,b], and differentiable on the open interval (a, b), then there exists some c ∈ (a,b), such that $$\frac{f'(c)}{g'(c)}= ...
5
votes
1answer
77 views

For what values of $x$ does the series $\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$ converge?

I have to study the values of $x$ for which $$\sum_{n=1}^\infty \frac{1}{(\ln x)^{\ln n}}$$ converges. First we say that we must have $x>0$. Then, I have started by rewriting the series as ...
3
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0answers
55 views

How much algebra is necessary to understand Rudin's “Real and Complex Analysis”?

I've been reading up on the finite element method, and the text many people recommend is The Mathematical Theory of Finite Element Methods by Brenner and Scott. As part of the background, the authors ...
0
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1answer
9 views

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ (Riemann integral)

Let $f_1 , f_2: I\mapsto \mathbb{R}$ bounded functions. Show that $L(f_1)+L(f_2)\leq L(f_1+f_2)$ where $L(F)$ is the supremum of the lower sums of the Riemann integral. I tried to by contradicction ...
0
votes
1answer
27 views

Why $N= max(2,\frac {2}{\epsilon})$ for $|a_n -L|<\epsilon $ convergence problem [on hold]

Using the proof development strategy used regarding the proposition (for all $\epsilon \in \mathbb{R}^+$ there exists an $N \in \mathbb{R}^+$ such that $|a_n - L| < \epsilon$ for all $n > N $) ...
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2answers
23 views

Are as constant but not constant random variables trivial sigma-algebra-measurable? Converse?

Are almost surely constant random variables trivial sigma-algebra-measurable? These links suggest no: http://at.yorku.ca/cgi-bin/bbqa?forum=homework_help_2004&task=show_msg&msg=1121.0001 ...
0
votes
1answer
23 views

Riemann sum and integral approximation error (Lipschitz function)

Given a function is Lipschitz continuous on [0,1] such that ∀ x , y ∈ [0 , 1] , | f(x) - f(y) | ≤ M | x - y | for M ∈ ℝ how would you prove: I tried to use the fact that we can find a point in ...
7
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4answers
40 views

For which values of $\alpha \in \mathbb{R}$, does the series $\sum_{n=1}^\infty n^\alpha(\sqrt{n+1} - 2 \sqrt{n} + \sqrt{n-1})$ converge?

How do I study for which values of $\alpha \in \mathbb{R}$ the following series converges? (I have some troubles because of the form [$\infty - \infty$] that arises when taking the limit.) ...
0
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1answer
27 views

Understanding density of irrational numbers and Archemedian property

From Density of irrationals I know this much of the proof of the density of irrational numbers "We know that $y-x>0$. By the Archimedean property, there exists a positive integer $n$ such ...
1
vote
1answer
19 views

Locality of tensors part of definition?

I am wondering whether linearity with respect to scalar functions $f \in C^{\infty}(M, \mathbb{R})$ is part of the definition of a tensor? Let me explain it by referring to the Riemann curvature ...
-1
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1answer
33 views

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact?

If $f:X \to [0,1]$ is an onto continuous closed map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again ...
3
votes
2answers
61 views

how can I show this integral diverges?

I want to show $E(T_a)=\infty$ $$E(T_a)=\int_0^{\infty}{{x|a|}\over\sqrt{2\pi}}x^{-3/2}e^{-a^2/x}dx$$ to show this I need to show this integral diverges. I know gamma function that $$\Gamma ...
1
vote
1answer
52 views

Convex function inequality for Euclidean norm: $\|(f(x_1),\cdots,f(x_n))\|_2\leq f(\|x\|_2)$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a positive, convex, continuous function such that $f(0)=0$. (If you wish you can also suppose $f$ to be monotone increasing.) I would like to prove or to ...
2
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1answer
33 views

Why is this flux zero?

I have the vector field $\vec{F}=(x^2+y+2+z^2, e^{x^2}+y^2, x+3)$ and $S$ the part of the spherical surface $\{x^2+y^2+(z-a)^2=4a^2\}$ that is above the $x,y$-plane, with orientation outwards. I know ...
2
votes
1answer
33 views

A piecewise $C^1$ curve has Jordan measure zero.

$\newcommand{\Reals}{\mathbb{R}}\gamma:[0,1]\to \Reals^2$ is an injective parametrization of a curve $\Gamma$, which is piecewise $C^1$ and the length of the curve is $L(\Gamma_k)<\infty$. 1.1.: ...
2
votes
1answer
39 views

How to tell if a series diverges or is indeterminate? Study of some cases of $\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n$

Suppose we have a series dependent on a parameter. For example: $$\sum_{n=1}^\infty3^n (1+\frac{1}{n})^{n^2}k^n.$$ By root test, we know that this series absolutely converges (hence converges) if ...
2
votes
1answer
29 views

differentiable and uniform continuity of f and F

Given $f: \Bbb R \to \Bbb R$. define new function: $F(x) =\frac{f(x)-f(a)}{x-a}$ for $x\neq a$. Prove that $f$ is differentiable at $a$ if and only if $F$ is uniformly continuous in some punctured ...
2
votes
2answers
57 views

If $f:X \to [0,1]$ be an onto continuous map and $\{f^{-1} (y)\}$ is compact then Is $X$ compact?

If $f:X \to [0,1]$ is an onto continuous map and $\{f^{-1} (y)\}$ is compact for every $y \in [0,1]$, then is X necessarily compact? Now continuous image of a compact set is compact. Again $X$ is ...
0
votes
0answers
23 views

Sums convergent but not uniformly convergent on [0,1]

Show that both $\sum_{n=1}^{\infty} ({1-x}){x^n}$ and $\sum_{n=1}^{\infty} (-1)^n({1-x}){x^n}$ are convergent on [0,1] but only one converges uniformly. Which one? Why? I was playing around with the ...
1
vote
1answer
37 views

How to prove a function is continuous on a compact set?

I´m struggleing with this problem: I know by theorems that inf(d(a,b)) exists if the real value function d is continuous on the set AxB. But how can I prove that d is continuous?
0
votes
1answer
21 views

Upper and lower bound on different of ${\rm erf}(\frac{x+c}{b})-{\rm erf}(\frac{x-c}{b})$

I am trying to find a good upper bound on \begin{align*} f(x)={\rm erf}\left(\frac{x+d}{b}\right)-{\rm erf} \left(\frac{x-d}{b}\right) \end{align*} here $d>0$ I know that $f(x)$ is symmetric ...
0
votes
2answers
39 views

is a convex continuous function absolutely continuous

Does a continuous convex function $\mathbb{R} \to \mathbb{R}$ belong to $W^{1,1}_{loc}$ ? thank you.
-3
votes
0answers
38 views

How to use Stokes theorem [on hold]

Can anybody help me solve this problem? Stokes theorem to get $\oint \vec{F}d\vec{R}$
-2
votes
1answer
36 views

Is p a limit point of the range of {$P_n$} when it converges to p [on hold]

Is $p$ a limit point of the range of {$P_n$} when it converges to p? https://www.youtube.com/watch?v=VEx3Ys6JAJo T/F question (d) in this video. Professor argued: p is a limit point of {$P_b$} but ...
0
votes
2answers
33 views

Proving Integral Test?

Assume that $f(x) \geq 0$ and that $f$ decreases monotonically on $[1, \infty]$. Prove $\int_{1}^{\infty} f(x)dx$ converges iff $\sum_{n=1}^{\infty} f(n)$ converges. My proof: If $f$ is non-negative ...
2
votes
3answers
100 views

Stokes theorem to get $\oint \vec{F}d\vec{R}$

I have the vector field \begin{equation*} \vec{F}=(ye^x,x^2+e^x,z^2e^z) \end{equation*} and the curve $C$ that us given by \begin{equation*} \vec{r}(t)=(1+\cos t, 1+\sin t, 1-\cos t-\sin ...
0
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0answers
20 views

Sequence of functions with compact support.

Let $(f_k)_{k \in \mathbb{N}}$ be a sequenz of functions with compact support from $\mathbb{R}^n$ to $\mathbb{R}$ and let $f$ be any function from $\mathbb{R}^n$ to $\mathbb{R}$. a) Define exactly, ...
0
votes
3answers
52 views

What does “all but” means? Rudin 3.2

Rudin thm 3.2 (a) $P_n$ converges to p iff every nbhd of $p$ contains ALL BUT finitely many of the terms of $P_n$
1
vote
1answer
19 views

Divergence theorem to calculate the flux

I have the vector field $\vec{F}(x,y,z)=(-x,-y,z^2)$ and i want to find the flux through the part of the cone $\{z=\sqrt{x^2+y^2}\}$ between the planes $z=1$ and $z=2$. How do I use the divergence ...
-4
votes
1answer
50 views

Direct proof: if $P_n \rightarrow p$ and $P_n \rightarrow p'$, then $p=p'$ [on hold]

Direct proof: if $P_n \rightarrow p$ and $P_n \rightarrow p'$, then $p=p'$
1
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0answers
30 views

Topological properties of regular and critical points and values

Let $f\colon M\rightarrow N$ be a smooth map between smooth manifolds. Consider the following two statements, the second one under the assumption The set of regular points of $f$ are open in $M$, ...
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0answers
34 views

Proofs involving 3 quantifiers: A(3,3)=6 cases [on hold]

I knows how to prove statement involving 1 or 2 quantifiers. So there are 6 combinations of 3 universal quantifiers ("for all" and "there exist") with an extra implication that makes a quantified ...
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0answers
21 views

Sequence converges to x0 iff inverse of f is continuous at f(x0) [on hold]

Let $A$ and $B$ be two nonempty subsets of $\mathbb R^n$ and $f:A\to B$ be a bijection. Show that $f^{-1}$ is continuous at a point $y_0=f(x_0)$ iff whenever $\|f(x_n)-f(x_0)\|\to0$, the sequence ...
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votes
2answers
33 views

Can there be a sequence which is not convergent but has accumulation points?

The sequences which are convergent are inevitably bounded. So, they have bound points and hence have accumulation points. But at one place, my book mentions that it is not necessary for a sequence ...
0
votes
2answers
43 views

$F(x) = \max(f(x),g(x))$ and $G(x) = \min(f(x),g(x))$. Prove that $F,G$ are continuous functions. [on hold]

Let $f,g:\mathbb{R}\to\mathbb{R}$ be continuous. $F(x) = \max(f(x),g(x))$ and $G(x) = \min(f(x),g(x))$. Prove that $F,G$ are continuous functions.
2
votes
2answers
30 views

Uniform convergence defined recursively

Define a sequence of functions $y_n(x)$ by $y_0(x)=0$, $y_n(x) = Ty_{n-1}(x)$, where $$Tf(x) = x^3 + \int_0^x t^3 f(t)dt$$ Where I've shown that $T:B \rightarrow B$ is a contraction in the Banach ...
0
votes
1answer
19 views

Lower semecontinuity at infinity

Let $f$ be an extended value function on $\mathbb R^n$ that is bounded from below, i.e., $$f(x)\geq\alpha, \; \forall x \in \mathbb R^n.$$ My question is: Is it true that $$\liminf_{\|x\|\to +\infty} ...
4
votes
2answers
48 views

Largest value of an unknown function evaluated at a particular x value

Firstly, apologies for the extremely vague title; the problem I'm working on doesn't particularly possess a specific title. I am trying the following question but am very stuck: Assume the function ...
-6
votes
1answer
59 views

Prove if f(x) = g(x) for every x in Q, then f = g [closed]

Prove if f(x) = g(x) for every x in Q, then f = g
3
votes
3answers
95 views

How to show that $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function

(This is a homework problem) I am trying to show that the series $\sum_{k=0}^{\infty} \frac{x^{k}}{k!}$ represents a continuous function on $\mathbb{R}$. My idea was to show that the functions ...
2
votes
2answers
35 views

Continuity and differentiability of the function $x|x|$

Let $f:\mathbb R \to \mathbb R$ defined by $f(x) = x|x|$, Is the function continous at all points? If it is, then is it differentiable at all points? Yes, the function is continuous everywhere but ...
-4
votes
1answer
33 views

real analysis diiferentiation [duplicate]

I have no idea how to approach this question. Please help. How do I use definition of improper integral to solve it?
4
votes
1answer
31 views

Finding a Radom-Nikodym derivative

Let $(X,\Sigma,\mu)$ be a measure space. Let $f_1,f_2\in L^1(\mu)$ and consider the signed measures $$v_i(E):=\int_Ef_id\mu$$ for every $E\in\Sigma$. If $v_1\ll v_2$ and $v_2\ll v_1$, we must find ...
3
votes
2answers
45 views

Proving $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$

Let a,b,x,y be positive reals. Prove $x\ln(\frac{x}{a})+y\ln(\frac{y}{b})\geq (x+y)\ln(\frac{x+y}{a+b})$ I don't have any olympic background, so I may be missing some standard trick. The ...
-2
votes
2answers
39 views

real analysis convergence and integration [on hold]

this question is asking you to prove the convergence of f. f is decreasing monotonically in [1 and f>-0. I have no idea how to approach this question. Please help:)
3
votes
1answer
35 views

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent:

Let $f: (X, d) \to (Y, d')$. Prove that the following are equivalent: a) $f$ is uniformly continuous in $X$. b)For every pair of sequences $(x_n), (y_n) \subseteq X$ such that $ d(x_n, ...