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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-2
votes
1answer
27 views

Does $\sum a_n$ converge if $a_1 = 1$ and $a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n$

Let $$a_1 = 1$$ and $$a_{n+1}=\frac{2+\cos n}{\sqrt n}a_n.$$ Consider $$\sum a_n.$$ How do I calculate if the series converges? The definition by recurrence troubles me a lot.
1
vote
0answers
21 views

$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for nonzero rational coefficients

is it possible to find $a,b\in\mathbb{Z}$ such that for every $k\in\mathbb{N}$ the sum $$ \sum_{n=1}^{\infty}\frac{1}{(an^2+b)^k} $$ equals $q_0+q_1\pi+\dots+q_k\pi^k$ for some nonzero ...
2
votes
2answers
39 views

Determine if a series defined by cases is convergent and calculate the sum

Consider $\sum_{n=1}^\infty a_n$, where $a_n$ is $$3^{-n}$$ if $n$ is even and $$\ln \frac{(n+2)(n+1)}{n(n+3)}$$ if $n$ is odd. I have to say if it is convergent and calculate its sum, but the ...
6
votes
1answer
45 views

the series: compute $ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $

Compute $$ \sum_{n=1}^{\infty}\frac{1}{(4n^2-1)^4} $$ the result is $\frac{\pi^4+30\pi^2-384}{768}$, so I'm sure the sums $\sum\frac{1}{n^2}$ and $\sum\frac{1}{n^4}$ should appear in the solution. ...
2
votes
1answer
23 views

If $\frac{n_k}{m_k} \rightarrow \xi \in \mathbb R \setminus \mathbb Q$ then $m_k \rightarrow \infty$

Is it true that if $\frac{n_k}{m_k} \rightarrow \xi \in \mathbb R \setminus \mathbb Q$ then $m_k \rightarrow \infty$? $m_k$ and $n_k$ are integers.
0
votes
0answers
13 views

Help in finding a paper on nonlinear Schrodinger equations

I'm looking for the following paper: Authors: Baillon, Jean-Bernard; Cazenave, Thierry; Figueira, Mario Title : Equation de Schrodinger non linéaire. (C. R. Acad. Sci. Paris Ser. A-B 284 ...
1
vote
2answers
54 views

Find $f$ so that $\int_{1}^{\infty}f(x)dx$ exists, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist? [duplicate]

I need help finding an example of a function such that $\int_{1}^{\infty}f(x)dx$ converges, but $\displaystyle \lim_{x\to\infty}f(x)$ does not exist. I was trying to find examples of functions ...
0
votes
0answers
22 views

Proof of $f(x+c) \rightarrow m$ as $x \rightarrow a$

If $f(x) \rightarrow m$ as $x \rightarrow a+c$ then $f(x+c) \rightarrow m$ as $x \rightarrow a$. Proof: Because of the assumption, we have $\forall \varepsilon >0$, there exists $\delta >0$ ...
1
vote
2answers
24 views

Is this an open set in $\mathbb R^2$?

Is $\{(x,y)\mid y = \sin \frac {1}{x}, x>0\}$ an open set? (It is living in $\mathbb R^2$.) I think it should be open because $(0,0)$ seems to be a limit point of this set while it is not an ...
4
votes
5answers
64 views

Calculate $\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$

How can I calculate the following limit? I was thinking of applying Cesaro's theorem, but I'm getting nowhere. What should I do? $$\lim_{n \to \infty} \ln \frac{n!^{\frac{1}{n}}}{n}$$
0
votes
0answers
10 views

Exactly one supporting line for a $C^1$ Jordan curve

Let $\gamma :[a,b]\to\mathbb{R}^2$ be a convex Jordan curve (closed, simple, continuous) that has $C^1$ regularity, with $\gamma '(t)\neq 0,\ \forall t\in [a,b]$. Prove that there is exactly one ...
0
votes
2answers
20 views

Continuous and bounded - Check my proof please

Let $f : [0, ∞) → \mathbb{R}$ be continuous such that $\lim_{x→+∞} f(x) = 0$. Prove that $f$ is bounded on $[0, ∞)$ By our hypothesis and the definition of continuity, given $ c \in [0, \infty), ...
1
vote
0answers
27 views

For which values of $x$ is the following series convergent: $\sum_0^\infty \frac{1}{n^x}\arctan\Bigl(\bigl(\frac{x-4}{x-1}\bigr)^n\Bigr)$

For which values of $x$ is the following series convergent? $$\sum_{n=1}^{\infty} \frac{1}{n^x}\arctan\Biggl(\biggl(\frac{x-4}{x-1}\biggr)^n\Biggr)$$
0
votes
0answers
15 views

Construction of the Ito integral

We fix some filtered probability space $(\Omega,\mathfrak{F},\{\mathfrak{F}_t\}_{t\in[0,T]},\mathbb{P})$. Let, for short, $L^2$ be the space of all progressively measurable processes in ...
2
votes
0answers
23 views

An exercise about Lebesgue measure

This is part of an exercise from "Real analysis for graduate students" by Richard Bass: Let $m$ be a Lebesgue measure. Suppose for each $n$, $A_n$ is a Lebesgue measurable subset of $[0,1]$. Let $B$ ...
1
vote
0answers
20 views

Is it Green formulation?

I have this expression: $$-\Delta_p v_n+|v_n|^{p-2}v_n-\theta_n |v_n|^{p^*-2}v_n\rightarrow 0 ~~\text{in}~~ (W^{1,p}_0(\Omega))'$$ It is sying that this expression imply that ...
5
votes
1answer
57 views

If $f(0)=f(1)=f(2)=0$, $\forall x, \exists c, f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$

Let $f:[0,2]\to \mathbb R$ be a $C^3$ function such that $f(0)=f(1)=f(2)=0$ Prove that $\forall x\in[0,2], \exists c\in[0,2], f(x)=\frac{1}{6}x(x-1)(x-2)f'''(c)$ This problem got me stuck. I ...
2
votes
1answer
29 views

Uniform Convergence of a Recursive Function

Let $f_{0}$ in $ \mathbb R $ be an increasing and continuous function. Define $f_{n}$ as $f_{n+1}=\arctan\left(x+f_{n}(x)\right)$ for $x$ in $\mathbb R$, $n \geq 1$. Show that $(f_{n})$ converges ...
1
vote
1answer
31 views

Uniform convergence of recursive sequence

I have the following exercise: Let $\varphi:[0,+\infty)\to \mathbb{R}$ an increasing continuous function that satisfies that $1/2\leq \varphi(x) <1$ for all $x>0$. Let $f_0:[0,+\infty)\to ...
2
votes
1answer
16 views

Pointwise convergence - $\frac{nx}{1+n \sin(x)}$ , $x \in [0, \frac{\pi}{2}]$

Is anyone able to check if this is correct: for $$f_n(x) = \frac{nx}{1+n \sin(x)} , x \in [0, \frac{\pi}{2}]$$ Does this converge pointwise to $$ \frac{x}{\sin(x)}$$ I am unsure due to the fact ...
1
vote
0answers
30 views

Prove that $g$ is differentiable and that $g'$ is not differentiable at $0$

For part ($c$) I assume I have to show that the limits from above and below are equal? I am having trouble doing this though... I get limit as $h$ tends to $0$ of ( $(4-h^2)^{0.5}$ - ($4^{0.5}$) ...
1
vote
1answer
19 views

find extremums of $x^2+y^2-12x+16y$ on compact set

I'm trying to find the max/min points of the function \begin{equation*}f(x,y)=x^2+y^2-12x+16y\end{equation*} on the set \begin{equation*}D=\{(x,y):x^2+y^2\leq1\space,\space 3x\geq -y\}\end{equation*} ...
0
votes
0answers
9 views

unique inner product on a tensor product of Hilbert $C^*$ modules and Hilbert spaces.

For a $C^*-$ algebra $A$ and a Hilbert space $H$ and a Hilbert $A-$module E; how can we show that there is a unique $A-$ valued inner product on $H \otimes E$ as $< h_1 \otimes x_1 , h_2 \otimes ...
0
votes
0answers
18 views

Prove that the function $ξ\in R \mapsto {e^{i\cdot ξ\cdot λ}-1\over i\cdot ξ}-λ$ is $C^{\infty}$ is $C^{\infty}$

Prove that the function $$ξ\in R \mapsto {e^{i\cdot ξ\cdot λ}-1\over i\cdot ξ}-λ$$ is $$C^{\infty}$$ (and in the point of ξ=o) Any ideas how to prove this? i am trying to think some ideas but i can ...
0
votes
0answers
17 views

multiplicative inverse of a medible function

I want to prove that if $f$ and $g$ are lebesgue measurable functions then $h$ defined by: $$ h(x):=\frac{f(x)-g(x)}{f(x)+g(x)} \text{ if } f(x)+g(x)\neq 0 \\ 0\text{ if } f(x)+g(x)= 0 $$ ...
0
votes
1answer
31 views

if g is not constant zero, $f\circ g$ has a local minimum at zero

Consider $f:\mathbb{R}^2\to\mathbb{R}\; f(x,y)=(x^2-y)(x^2-3y)$ and a linear function $g:\mathbb{R}\to\mathbb{R}^2,\; x\mapsto \begin{pmatrix} g_x(x)\\ g_y(x) \end{pmatrix} $. The claim is: If $g$ ...
0
votes
1answer
30 views

What does this notation mean? Functional Analysis

I am studying analysis at the moment and came across this notation and I would like to know what it really means: $$C_{c}^{\infty}(\Omega)$$ My understanding so far is that,this is the space of ...
0
votes
0answers
31 views

Exercise to Heat equation

I am dealing with the following exercise on a generalized heat equation Let $U$ be a open bounded subset of $\mathbb{R}^n$ and $T \in (0, \infty).$ Now, let $u$ be a solution to $$(\partial_t - ...
0
votes
2answers
23 views

Recurrence sequences with two initial condition: how do I calculate the limit?

I've done some exercises with recurrence sequences with one initial condition. So, now that I'm attempting one exercise with two initial conditions I'm confused. Could you show me what to do? Let ...
0
votes
1answer
42 views

Limit of $a_{n+1}= \frac{n}{n+1} a_n$

I think that this sequence $$a_{n+1}= \frac{n}{n+1} a_n$$ can be rewritten as $$a_n= \frac{1}{n+1}a_0.$$ Therefore the limit should be $0$. But my proof by induction turns out wrong. Is my idea ...
1
vote
2answers
30 views

Counter example in Sobolev spaces: Is there a standard example for showing that $H^1(\mathbb{R}^2)\not\subset L^\infty(\mathbb R^2)$?

One way to define the Sobolev space $H^s(\mathbb{R}^n)$ is as the completion of the $C^\infty$ functions on $\mathbb{R}^n$ under the norm $$\lVert f \rVert_{H^s(\mathbb{R}^n)}:= \left( ...
0
votes
1answer
35 views

Bring a proof for the fundamental theorem of calculus.

If $f\in \mathscr{R}$ on $[a,b]$ and if there is a differentiable function $F$ on $[a,b]$ such that $F'=f$, then $$\int_a^b f(x)\ \ d(x)=F(b)-F(a)$$
1
vote
4answers
66 views

Find the limit of the sequences: $a_{n+1}=3a_n - n + 1$ and $(a_n)^\frac{1}{n}$ with $a_0 > 0 $

Let $a_0 > 0 $ and $$a_{n+1}=3a_n - n + 1.$$ I have to find its limit. I have also to find the limit of $(a_n)^\frac{1}{n}$. But this seems even more complicated. For the first part I've used the ...
1
vote
1answer
24 views

Convergence of $u * \eta_\epsilon$

Let $\eta \in C_c^\infty(B(0,1)), \eta \ge 0, \eta$ radially symmetric and $\int_{\mathbb{R}^n} \eta d\mathcal{L}^n = 1$. $\eta_r := r^{-n} \eta(\frac{x}{r}) \in C_c^\infty(B(0,r))$. Integral of ...
2
votes
3answers
59 views

Prove that a certain sequence is increasing and find its limit: $a_1 = 1$ and $a_{n+1}=n(1+\ln a_n)$ (and $(a_n)^\frac{1}{n}$)

Let $a_1 = 1$ and $$a_{n+1}=n(1+\ln a_n).$$ I have to find its limit. I want to prove that it is increasing for starters, but I'm already stuck. What should I do? I have also to find the limit of ...
0
votes
1answer
21 views

Little o notation within another little o

To prove $e^{x + o (x)} = 1 + x$ as $x \rightarrow 0$, I can do it directly: $\lim_{x \rightarrow 0} \frac{\log (1 + x) - x}{x} \overset{\text{l'hopital}}{=}\lim_{x \rightarrow 0} \frac{(1 + x)^{- ...
2
votes
1answer
63 views

A necessary condition to $F'(x)=f(x)$ for a continuous function $f$

Theorem: Consider , $$F(x)=\int_a^xf(t)\,dt$$ If the function $f:[a,b]\to \mathbb R$ is continuous then $F(x)$ is differentiable and $F'(x)=f(x).$ I know that the continuity condition ...
-2
votes
0answers
29 views

How can the elements $a_1, a_2, a_3\ldots, a_n$ be distinct in Theorem 2.13 of Rudin?

In Theorem 2.13 of Rudin, how could the elements $a_1,a_2,\ldots, a_n$ be distinct like he says they can? $A$ is a countable set (or just a set) and, therefore, all elements must be distinct. Perhaps ...
2
votes
1answer
23 views

Radius of convergence of a power series with $a_n$ convergent

Let $\{a_n:n \geq 1\}$ be a sequence of real numbers such that $\sum_{n=1}^{\infty} a_n$ is convergent and $\sum_{n=1}^{\infty} |a_n|$ is divergent. Let $R$ be the radius of convergence of the power ...
1
vote
1answer
15 views

If $X=\{0,1\}$, there exists an outer measure $\mu^*$ on $X$ such that $\mu^* \neq \mu^+$

Background Let $\mu^*$ be an outer measure on $X$ , $\mathcal{M}^*$ the $\sigma-$ algebra of all $\mu^*$ measurable sets, $\overline{\mu}=\mu^*\bigg|_{\mathcal{M}^*},$ and $\mu^+$ the outer measure ...
0
votes
1answer
28 views

If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$.

TRUE or FALSE: If $f$ is a convex function in $(a,b)$ then $|f|$ is convex function in $(a,b)$. My Proof: Since $f$ is convex function so, $f(tx+(1-t)y)\le tf(x)+(1-t)f(y)$ , for all ...
1
vote
1answer
34 views

Explanation of taylor series

I understand that for a Taylor series of a function $f(x)$, centered around the point a, the general expression can be written as: $$ \begin{align} &f(x) \\ &= f(a) + f'(a) (x-a) + ...
1
vote
1answer
37 views

Cauchy Sequences To Prove $f(z)$ is not continous [on hold]

I've learn ways to prove the discontinouity of a complex function. I have not learn Cauchy Sequences however. I cannot find useful information on the subject. Please explain
4
votes
0answers
39 views

Positivity of alternating series

Let $\{a_n\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $\limsup_n \frac{1}{n}\log a_n=-\infty$. Then $$ f(x)=\sum_{n\geq 0}a_n x^n $$ converges absolutely for all $x$. Under ...
1
vote
0answers
15 views

Analytic function with alternating taylor series

Is there a characterization of the sequences of non-negative real numbers $\{a_n\}_{n=0}^{\infty}$ such that $$ f(x)=\sum_{n\geq 0}a_n (-x)^n $$ converges absolutely for all $x\geq 0$? It's not hard ...
2
votes
4answers
95 views

Compute $\lim_{n \to \infty} \left(\frac{1}{\sqrt{n^3+1}} + \frac{1}{\sqrt{n^3+4}} + \cdots + \frac{1}{\sqrt{n^3+n^2}}\right)$

How do I evaluate the following limit? I guess I should do a comparison, but I've got no clue about what to do. Could you give me a hand? $$\lim_{n \to \infty}\left( \frac{1}{\sqrt{n^3+1}} + ...
-1
votes
1answer
43 views

About summer course or online course of Linear algebra and real anyasis [on hold]

I just looking for the online course for Linear algebra or real analysis but it should be upper level. i saw MIT and another college but our university said it was not upper level its likely ...
0
votes
2answers
41 views

Calculate: $\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$ and $\lim_{n \to \infty} \frac{10^{\sqrt{(\ln n)^2 + \ln n^2}}}{n^2+1}$

I have to evaluate the following limits (which are similar). However, I don't know how to evaluate them. Could you give me a hand? $$\lim_{n \to \infty} \frac{2^{\sqrt{(\ln n)^2 + \ln ...
5
votes
3answers
109 views

How can I calculate $\lim_{n \to \infty} (1 + \frac{1}{n!})^n$ and $\lim_{n \to \infty} (1 + \frac{1}{n!})^{n^n}$?

How do you calculate the following limits? $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^n$$ $$\lim_{n \to \infty} \left(1 + \frac{1}{n!}\right)^{n^n}.$$ I really don't have any clue ...
2
votes
1answer
45 views

Verify solution to ODE

I am given the ODE $$\left(f''(x)+\frac{f'(x)}{x} \right) \left(1+f'(x)^2 \right) = f'(x)^2f''(x)$$ and I already know that the solution to this ODE is given by $$f(x)= c \cdot arcosh \left( ...