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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40 views

On the sequence of function $f_n(x)=n^2x(1-x^2)^n $

Is the sequence of functions $f_n(x)=n^2x(1-x^2)^n $ uniformly convergent over $[0,1]$ ? Is the series $\sum_{n=1}^{\infty} f_n(x)$ convergent with $0<x<1$ ? Is the series uniformly convergent ...
0
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0answers
20 views

Proof that the sum of a certain infinite series can be bounded to zero

$\forall 0 < \alpha < 1$, there exists $\lambda > 0$, $k > 0$, s.t. $$ \lim_{n \to \infty} \sum_{w = 1}^{\lambda n} \binom{n}{w} \frac{1}{2^{\alpha n}}\left(1 +\left(1 - ...
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0answers
20 views

Continuity and Supremum

Attempt: I can't seem to gauge the points of continuity, for the attain supremum parts, I know I need to use the fact a continuous function on a closed, bounded interval is bounded and attains its ...
4
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2answers
49 views

Problem 7 IMC 2015 - Integral and Limit

I'm trying to solve problem 7 from the IMC 2015, Blagoevgrad, Bulgaria (Day 2, July 30). Here is the problem Compute $$\large\lim_{A\to\infty}\frac{1}{A}\int_1^A A^\frac{1}{x}\,\mathrm dx$$ ...
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1answer
37 views

Can functions with a non-analytic point always be approximated with power laws around the special point?

I'm interested in continuous functions from $\mathbb{R}^n$ to $\mathbb{R}$ that fail to be analytic at a given point (let's say the origin), while still being analytic in a region surrounding it. ...
3
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2answers
83 views

pointwise almost everywhere convergent subsequence of $\{\sin (nx)\}$

Can you prove or disprove that the sequence $\{\sin (nx)\}$ has a pointwise convergent almost everywhere subsequence with respect to the Lebesgue measure on $\mathbb{R}$ ? Edit: I am adding my ...
2
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1answer
22 views

Distance of two hyperbolic lines

Consider the upper-half plane model of the hyperbolic plane $\mathbb {H}^2.$ Now consider two lines in it given as $\ell_1:=\lbrace { (x, y)\in \mathbb {H}^2 \vert x^2 +y^2=r^2\rbrace}, ...
2
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4answers
51 views

Elegant solution for $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$

I have the following integral: $\int {\frac{\cos(y)}{\sin^2(y)+\sin(y)-6}}dy$ I already know the solution, but it needs three substitutions. Is there a simpler, more elegant way to go about this?
4
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1answer
57 views

About the common exercise using Weierstrass Theorem

A common exercise using Weierstrass Theorem is: If a continuous function $f: [0,1] \rightarrow \mathbb{R}$ satisfies: $$\int _{[0,1]}x^kf(x)dx=0$$ for all $k \in \mathbb{N}$, then ...
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2answers
51 views

Does l'Hopital rule work for -inf/inf?

If you have an indeterminate form: $\frac{-\infty}\infty$ $\frac\infty{-\infty}$ $\frac{-\infty}{-\infty}$ does l'Hôpital's rule also apply?
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0answers
12 views

$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n)$

Do you have a reference for the following intuitive result? Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequence of reals. Then $$\liminf_n \min(a_n,b_n)=\min( \liminf_n a_n, \liminf_n b_n).$$
3
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2answers
35 views

Extend absolutely continuous function to be bounded and continuous on $[0,1]$

So I am working on the following problem and I am not really sure how to go about approaching it. I always seem to get stuck when trying to start a problem involving the extension of a function to the ...
18
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3answers
552 views

About the integral $\int_{-1}^1 \frac{1}{\pi^2+(2 \operatorname{arctanh}(x))^2} \, dx=\frac{1}{6} $

Here is a question that naturally arose in the study of some specific integrals. I'm curious if for such integrals are known nice real analysis tools for calculating them (including here all possible ...
4
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2answers
83 views
+300

Proving the continuity of these maps

Backstory: I am having an exam soon, and these are the assignments that keep coming up, I cannot finish any of them to the end, but have ideas about solving them, and would like to hear your thoughts ...
3
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1answer
60 views

How to resolve the issue of two sequences converging to zero for $n, m \to \infty$?

My question is motivated by the following exercise in probability theory: Let $X_n \to X$ in probability and $X_n \geq Y$ a.s. Show that $X \geq Y$ a.s. I noticed that for all $n, m \in ...
11
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2answers
297 views

Find a closed form of the power series

Let a power series $$S(x)=\sum_{n=1}^{\infty}\frac{x^{n}}{4n+1},$$ then $1$ is the radius of convergence of $S$ .In fact $S(x)$ convergens for each $x\in[-1,1).$ My work is to find a closed form of ...
11
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3answers
749 views

Differentiable at a point

My roommates and I have an argument you guys can help to settle (peace is at stake, don't let us down!) In undergrad calculus courses, one usually explains what it means for a function to be ...
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1answer
33 views

Tough problem on sum of infinite series

I've been working on the problem for quite a while but have no idea how to approach it. This proposition arises from a practical probabilistic bound problem, but it seems very deep. Lots of thanks to ...
5
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4answers
68 views

Asymptotic of Inverse Function

Suppose we choose a positive constant $c$ and let $f_c(x)=\frac12x^2+cx^{3/2}$. I would like to get an asymptotic estimate for the function $f_c^{-1}(x)$ as $x\rightarrow\infty$. I assume it will be ...
3
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2answers
66 views

Geometry of images of maps $f: \mathbb R \to \mathbb C$?

I am having trouble seeing what a continuous map $f: \mathbb R \to \mathbb C$ might look like. If it was linear it would look like a line but it's not clear to me what happens if it's any map. I ...
4
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1answer
64 views

Integral written as the integral of a measure

Let $(X,\mathcal M,\mu)$ be a measure space and let $f\in L^1(X,\mu)$ be a positive function. Show that $$\int_X f \, d\mu=\int_{(0,\infty)} \mu(\{f>t\}) \, dt.$$
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0answers
16 views

curl-free, conservative vector fields in complex analysis

I just verified that for the conjugate of an analytic function $\bar{f}$=u-iv, this conjugate function is curl-free - the Cauchy-Riemann equations forces this to be the case. Then we can consider ...
2
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1answer
433 views

Approximating integrals with step functions

For $f \colon [1,2] \to \mathbb{R}$ , $f(x) = 1/x$, Choose a sequence of step functions $\phi_n$ approximating $f$ with partition $P_n := [r/n : n < r < 2n]$ to show that $ 1/(n+1) + \cdots + ...
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1answer
34 views

Is that form of Cesàro's theorem correct?

First, for sequences, we know that : "If a sequence $(a_n)_{n \ge 1}$ converges to $l\in \mathbb{R}$, then the sequence $(b_n)_{n \ge 1}$ defined by : $b_n=\frac{1}{n} \sum \limits_{k=1}^{n}a_k$ ...
2
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2answers
59 views

$m(E)=0$ or $m(E^c)=0$

The question comes from former qualifying exam of the graduate school I'm going to attend. Q: Suppose $E$ is measurable and $E=E+\frac{1}{n}$ for every natural number $n\geq 1$. Show that either ...
2
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1answer
52 views

Difference between line integrals in complex analysis and real analysis,

The formula in complex analysis is $$\int f(\gamma(t))\cdot(\gamma'(t)dt$$ and the formula in the real variable setting, for a gradient field, is: $$\int F\cdot dr$$ $$=\int f_x\,dx + f_y\,dy + ...
3
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1answer
58 views

What is uniform continuity geometrically

I just wonder if someone can give me clear picture because I do understand the technicalities but I still don't have a clear picture of the difference in the geometry between a uniform continuous ...
122
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2answers
8k views

Evaluate $ \int_{0}^{\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx$

Evaluate the following integral$$\int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan x)}\,\mathrm dx$$ My Attempt: Let $$\tag1 I = \int_{0}^{\Large\frac{\pi}{2}}\frac{1}{(1+x^2)(1+\tan ...
6
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1answer
56 views

Convergence of $n^{-\gamma}T$ where $T$ a hitting time for uniform rvs, can I use CLT?

Let $X_1,X_2,\dots$ be iid uniform on $\{1,\dots,n\}$ and define $T=\inf\{k:X_k=X_r \text{ for some }r<k\}$. The objective is to figure out when $n^{-\gamma} T$ converges weakly to some ...
0
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2answers
101 views

Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives.

Let U be an open subset of $\mathbb{R}^n$ and C a compact subset of U. Suppose that $f : U \mapsto \mathbb{R}$ has continuous first partial derivatives. Prove that f is Lipschitz on C. Thoughts: Let ...
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4answers
92 views

Are there an infinite number of open balls in an open set in a metric space?

Let's start off by recalling the definition of an open set in a metric space: A set $A$ in a metric space $(X,d)$ is open if for each point $x\in A$ there is a number $r\gt0$ such that $B_r(x)\subset ...
9
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0answers
132 views
+50

Linear differential equations of the $n$th order

$$ L(x)=x^{(n)}+a_1(t)x^{(n-1)}+\cdots +a_{n-1}(t)x'+a_n(t)x;\qquad a_1(t),a_2(t),\ldots\in C$$ $$U_j(\varphi)= \sum_{k=0}^{n-1}(M_{jk} \varphi^{k}(\alpha)-N_{jk} \varphi^{k}(\beta))= \gamma_j\quad ...
1
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2answers
46 views

What is a real world example of “zero work” done by a conservative vector field?

I have only a high school physics background, so when I study the later parts of multivariable calculus, e.g., Greens, Gauss, and Stokes' theorems, there are some topics that I only know the ...
6
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0answers
90 views

Wanted: Simple integration theory

Supposing we want to formulate a very primitive theory of integration, the only requirement being that all continuous functions $[a, b]\longrightarrow\mathbb{R}$ be integrable. What is the simplest ...
0
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3answers
48 views

Continuous but not uniformly continuous example

Let $f(x) = \frac{1}{x}$ for $x > 0$ and take our set at which the function act on $(0,1]$. This function is continuous but not uniformly continuous on $A$. To prove this consider $\epsilon = ...
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2answers
30 views

Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$

My question is, Is it true that $x_k\rightarrow x$ iff. $\exists N \in \Bbb{N}$ st. $k>N$ implies $|x_k-x|<a_k$ for some $a_k$ where $a_k>0$ and $a_k \rightarrow 0$ as $k \rightarrow 0$. ...
1
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1answer
327 views

Hilbert cube is compact

Let $\{u_n\}_{n\in \mathbb N}$ be an orthonormal set in $H$ (Hilbert space). How prove that the set $\displaystyle Q=\{x\in H :\ x=\sum_{i=1}^{\infty}{c_nu_n}, \ \mbox{where} |c_n|\leq\frac{1}{n} \}$ ...
2
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1answer
25 views

Minimizing long equation with hyperbolic functions

In physics book that I am reading it is said that minimizing the expression $$\phi = - N T k \log (2 \cosh(H \beta)) - \frac{J N}{2} z \tanh^2(H \beta) + H N \tanh(H \beta) $$ with respect to $H$ ...
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2answers
175 views

Proving a sequence converges when combinations of consecutive terms converge

Problem: Let $\{x_n\}$ be a sequence of real numbers such that $$\lim_{n\to\infty} 2x_{n+1}-x_n=L \in \mathbf{R}.$$ Prove that $x_n \to L$ as $n\to\infty$. I can see that if $\{x_n\}$ converges to a ...
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1answer
31 views

Integral upper bound

Let $A$ be a measurable set and $f$ an integrable function onto $[0,100]$ for example. Having knowledge of the value $\frac{\int_A f d\mu}{\mu(A)}$ (which in some sense is the average value of $f$) I ...
2
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1answer
38 views

Proving that a polilinear operator is differentiable

A Polilinear map operator is $P:X^1 \times ... \times X^n \to Y$ such that the foolowing applies: $\lambda, \mu \in R$ $$ P( \lambda x_1^1 + \mu x_2 ^1, x^2,...,x^n)= \lambda P(x_1^1,x^2,...x^n)+ \mu ...
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0answers
25 views

Is the limit of absolutely uniformly convergent series of functions a uniformly continuous function?

Let $f_n$ be a series of continuous and bounded functions on $\mathbb R$ such that $\sum|f_n|$ is uniformly convergent. Does $\sum |f_n|$ define a uniformly continuous function, and if so, how to show ...
3
votes
1answer
35 views

Convexity increases the “cost” of long steps

Let $V(n)$ be a non-decreasing, convex function on $\mathbb{N}$ such that $V(0)=0$, $V(1)=1$. Let $(r_i)_{i=1}^{N}$ and $(r^{\prime}_i)_{i=1}^{N^{\prime}}$, $N^{\prime} > N$, be two sequences of ...
0
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1answer
26 views

Question about the definition of convergence in measure.

In my text book convergence in measure ${{f}_{k}}\mathop \to \limits^{m} {f}$ is defined as "$\forall \epsilon> 0$ we have $\mathop {\lim }\limits_{k \to \infty } |\{ x \in \Omega :\left| ...
1
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2answers
47 views

Relation between bounded derivative and limit of a function

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuously differentiable real function such that $f(0) = 0$ $f'(x) \le -\frac{1}{2}$ for every $x \in \mathbb{R}$ Then it is ...
0
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1answer
24 views

Strange inequality in the proof of differentiability of Fourier series

I am looking at a proof and I found a strange inequality. Let $n\in \mathbb{Z}^d$ then it is stated that $\sum_{j=1}^d{(2\pi)^{2k}n_j^{2k}}>>\parallel n\parallel_2^{2k}$ due to the inequality ...
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3answers
36 views

Why there is a unique $c$ s.t. $f(c)=0$?

Let $f:\mathbb R\to\mathbb R$ derivable such that $f'(x)\geq 7$ for all $x\in\mathbb R$. Show that there is a unique $c\in\mathbb R$ s.t. $f(c)=0$. My try The unicity of such a $c$ is clear by ...
0
votes
1answer
17 views

Do all continuous piecewise affine functions belongs to the class ($A_1$) of Muckenhoupt functions?

Given $\Omega\subset\mathbb R^N$ is open, we say $\omega$: $\Omega\to [0,+\infty)$ belongs to Muckenhoupt class $A_1$ if there exists some $C>0$ such that $$ \frac{1}{|{B}|}\int_{B(x,r)} ...
1
vote
1answer
28 views

Uniform convergence and continuous.

Let $(f_{n})$ be a sequence of functions. Is it possible that $(f_{n})$ converges uniformly where each functions (that is $f_{1},f_{2}, f_{3}\dots$) aren't necessarily continuous?
1
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2answers
29 views

If $f$ is Lipschitz continuous on a closed interval $[a,b]$ such that $f([a,b])\subseteq [a,b]$ then it has a unique fixed-point

I am stucked at this problem: Prove or give a counter-example for the following sentence: If $f:[a,b]\to\Bbb{R}$ is Lipschitz continuous on a closed interval $[a,b]$ and $f([a,b])\subseteq [a,b]$ ...