2
votes
1answer
18 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
1
vote
1answer
39 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
1
vote
1answer
17 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
0
votes
0answers
26 views

Find an analytic function [duplicate]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
0
votes
2answers
17 views

A question about series ratio test

Could you please give me some hint how to deal with this question: Suppose $\left|\frac {a_{n+1}}{a_n}\right|\le c_n$ for each n and $c_n<1$. May we conclude that $\left|\frac ...
0
votes
0answers
37 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
5
votes
1answer
44 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
1
vote
3answers
42 views

Convergence of $\sum \frac{\sqrt{a_n}}{n^p}$

For $a_n \geq 0$, and $\sum a_n$ convergent, show that $\sum \frac{\sqrt{a_n}}{n^p}$ is also convergent for $p > 1/2$? What bugs me more is why isn't $\sum \sqrt{\frac{a_n}{n}}$ convergent?? ...
0
votes
1answer
28 views

A question about limsup and limif

Could you please help me understand this question: Suppose $a_n$ is bounded sequence and $A<\liminf a_n$, $B>\limsup a_n$. Prove : $A<a_n<B$ for all n>N. It seems to me to simple to be ...
0
votes
0answers
17 views

Does this recursion/sequence of iterated infinite sums converge?

Let $n,x\in\mathbb{N}$, $\alpha,\beta,\lambda\in\mathbb{R}^+$, where $\alpha,\beta<1$. Does the following sequence converge (and to what)? $s_0=\alpha n+\lambda$ ...
3
votes
3answers
119 views

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges

determine if $\sum^{\infty}_{n=1} \frac{n}{e^n}$ converges. I used the ratio test and eventually came to that: $\frac{n+1}{en}$ which is approximatley equal to $\frac{1}{e}$ which is less than 1. ...
0
votes
0answers
37 views

Bound on $f_n'$ implies uniform convergence of $f_n$?

Let $f_n$ be a sequence of functions that converge pointwise to a function $f$. Suppose I know that $|f_n'(x)| \leq C(x)$ where the constant doesn't depend on $n$. How do I conclude that $f_n \to f$ ...
-1
votes
0answers
22 views

An example of a sequence of continuous real functions pointwise convergent, but nowhere locally uniformly convergent? [duplicate]

I've been trying to come up with an example of a sequence of continuous real function which would converge pointwise everywhere, but nowhere converge locally uniformly, but I can't really think of ...
2
votes
2answers
53 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
1
vote
3answers
43 views

Need to know how: $L^{p}$ convergence for function.

I feel terrible at the moment as I have exhausted everything possible to understand this and I am still stuck. I have looked everywhere for some sort of resource to the concept at hand, and yet, they ...
1
vote
1answer
29 views

Uniform convergence and differentiation proof of remark 7.17 in Rudin's mathematical analysis

Rudin page 152 Theorem 7.17: Suppose $\{f_n\}$ a sequence of functions, differentiable on $[a,b]$ and such that $\{f_n(x_0)\}$ converges for some point x0 on [a,b]. If {f′n} converges uniformly on ...
4
votes
1answer
56 views

Different types of Convergence for a Series Function

I am currently investigating the convergence of the following function, $f(x)=\sum\limits_{k=1}^{\infty} \dfrac{x^{k}+\sin(k)}{k^{2}}$ for different "senses". I have shown that $f(x)$ converges ...
1
vote
1answer
39 views

How to prove : If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges.

Could you give me some hint how to prove this statement: If $a_{2n},a_{2n-1},a_{7n}$ converges than $a_n$ also converges. I think, obviously wrong, that if $a_{2n}\to a$ and $a_{2n-1}\to b$ than ...
2
votes
3answers
83 views

How to read $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$

Can somebody explain me step by step why does this $\bigcap\limits_{r=1}^{\infty}\bigcup\limits_{k=1}^{\infty}\bigcap\limits_{n,m\ge k}^{\infty}\{x:|f_n-f_m|<\frac{1}{r}\}$ represents the set of ...
3
votes
3answers
87 views

Tough Limit Question

Let {$x_n$} be monotone increasing sequence of positive real numbers. Show that if {$x_n$} is unbounded, then $\sum_{n=1}^{\infty}(1-\frac{x_n}{x_{n+1}})$ diverges.
2
votes
1answer
28 views

Big O notation preserved under convex functions?

Suppose that the random variable $X_T$ is $O_p(1)$ as $T \rightarrow \infty$, i.e. $\forall \epsilon>0$, $\exists M_\epsilon>0$ such that $\mathbb{P}(X_T>M_\epsilon)<\epsilon$ $\forall T$. ...
0
votes
1answer
24 views

Question about bounded sequence with two sub-sequential limits.

Could you please give me some hint how to deal with this question. Suppose $(a_n)$ is bounded sequence with 2 sub-sequential limits. Prove : there are real numbers A and B that ...
0
votes
1answer
24 views

weak convergence implies point-wise convergence?

If we have a bounded sequence $\{f_n\} \in L^p[a,b]$ that converges weakly to $f$ does this mean that the converges is also pointwise?? thank you.
2
votes
2answers
85 views

If $\sum a_n$ converges then $\sum (-1)^n \frac {a_n}{1+a_n^2}$ converges?

Could you please give me some hint how to deal with this question: If $\sum a_n$ converges, does this necessarily mean that $\sum (-1)^n \frac {a_n}{1+a_n^2}$ must converge also ? Thanks.
0
votes
0answers
28 views

Prove that $g_n \to g$ uniformly on $\mathbb{R}$ but $f_{n}g_n$ foes not converge uniformly to$fg$ on $\mathbb{R}$

Let $f_n(x) = x$ and $f(x) = x$ for $x \in \mathbb{R}$ and let $g_n(x) = \frac{1}{n}$ and $g(x) = 0$ for $x \in \mathbb{R}$. Prove that $f_n \to f$ uniformly on $\mathbb{R}$ and $g_n \to g$ uniformly ...
0
votes
0answers
26 views

Continuity of convergence vector

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i := ...
1
vote
0answers
26 views

Weak convergence implies uniform convergence in distribution?

Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow ...
1
vote
1answer
43 views

Does the sequence converge? To what?

Let α and β be positive real numbers and define a sequence by setting $s_1 = \alpha, s_2 = \beta$ and $s_{n+2} = \frac12(s_n+s_{n+1})\forall n\in \Bbb \ge1$ Show that the subsequences $\{s_{2n}\}$ ...
1
vote
1answer
52 views

Convergence of series with sum

Consider a continuous mapping $f: \mathbb{R}^n \rightarrow \mathcal{K} \subset \mathbb{R}^n$, where $\mathcal{K}$ is a compact convex set. Let $\alpha \in [0,1]$ and, for all $k \geq 0$, define $a_i ...
5
votes
0answers
55 views

Forcing series convergence

The following problem was posed by one of my lecturers: $(z_n)$ a null sequence in $\mathbb{C}$. Does there exist $(\epsilon_n)$ with each $\epsilon_n=\pm 1$ such that: $$\sum_n \epsilon_n ...
0
votes
2answers
21 views

Show convergence of improper integral with nearest integer function

Suppose $f$ is decreasing and $\lim_{x\rightarrow\infty}f(x)=0$. Then why $$\int_0^\infty(-1)^{[x]}f(x)dx$$ converges? ($[x]$ is the nearest integer function). Any hint?
2
votes
2answers
60 views

How to see this improper integral diverges?

$$\int^\infty_1\frac{1}{x^{1+1/x}}dx$$ I'm preparing for exams. I would also like to know what are some commonly used methods to show an improper integral diverges?
2
votes
4answers
96 views

Does this improper integral converge?

$$\int^\infty_0\cos x^3dx$$ I think no, because $\cos x^3$ keeps jumping between $-1$ and $1$. How to justify this rigorously?
0
votes
1answer
23 views

Show convergence of improper integral

Suppose $f(x)>0$ and $f$ is continuous on $[0,\infty)$ and $$\lim_{x\rightarrow\infty}\frac{f(x+1)}{f(x)}<1$$ How to see that $\int^\infty_0f(x)dx$ converges? I think I should use definition. ...
3
votes
1answer
43 views

Alternating series limit question [closed]

Suppose $b_n > 0$ for all $n\geq1$ and $$\lim_{n\to\infty}n\left(\frac{b_n}{b_{n+1}}-1\right)>0,$$ show that the alternating series $\sum_{n=1}^\infty(-1)^n b_n$ converges.
0
votes
0answers
92 views

understand proof of compactness in product topology

I am trying to understand the following reasoning. Call $\mathcal{F_\lambda}$ the set of functions $a:\mathbb{N} \to \mathbb{R}$ for which $Na(i) := \sum_{j \in \mathbb{N}} n_{ij} a(j)\leq \lambda ...
1
vote
0answers
33 views

Pointwise convergence to $\ln(x)$

I came up with a stepfunction on $(0,1]$: $s_n = \sum_{i=1}^{n} \ln (\frac{i}{n}) \chi_{(\frac{i-1}{n},\frac{i}{n}]} $, where $\chi$ denotes the characteristic function. I need to show that this ...
0
votes
1answer
45 views

Determine whether the series sin^2(1/n) converges or diverges

Determine whether the series sin^2(1/n) converges or diverges. Having real trouble with this one, I know all the terms are positive because it is being squared but I don't know where to begin with ...
1
vote
1answer
19 views

Convergence of two unusual “nested” sums

I was contemplating convergent sums, trying to think of very unusual or unorthodox sums that might be treatable recursively. Eventually, the following sum occurred to me: $$ \xi = 1 + \frac{ ...
0
votes
1answer
18 views

If a sequence of continuous functions converges pointwise to a continuous function on $ [a,b] $, it converges uniformly

If a sequence of continuous functions converges pointwise to a continuous function on $ [a,b] $, it converges uniformly. Looking at other theorems on the relationship between continuity and uniform ...
1
vote
2answers
35 views

Sequence with an infinite amount of limit points

Find a sequence which has an infinite amount of limit points. I was thinking about using the bijective pairing function $\langle\cdot,\cdot\rangle:\Bbb N\times\Bbb N\to\Bbb N,\langle ...
1
vote
2answers
93 views

Prove $x_n$ converges if $x_n$ is a real sequence and $s_n=\frac{x_0+x_1+\cdots+x_n}{n+1}$

Given that $x_n$ is a real sequence, $s_n = \frac{x_0+x_1+\cdots+x_n}{n+1}$ and $s_n$ converges, $a_n = x_n-x_{n-1}$, $na_n$ converges to 0, and $x_n-s_n=\frac{1}{n+1}\sum_{i=1}^n ia_i$, prove $x_n$ ...
1
vote
1answer
23 views

Conditions for convergences of a net

I got stuck on this problem and got no clue to solve it. Can anyone one here help me? I really appreciate. Let $X$ be a set and $\mathcal{A}$ the collection of all finite subsets of $X$, ...
3
votes
5answers
388 views

What is the difference between Cauchy and convergent sequence?

I am really confused. I will appreciate if somebody can help me to define the difference between Cauchy and convergent sequence. Many thanks.
-1
votes
2answers
39 views

Real Analysis Sequence convergence help

Suppose that $\{ y_n \}$ is a sequence of real numbers. Then $y_n$ approaches infinity as $n$ approaches infinity means that for every $M$ in the real numbers, there exists $N$ in the integers such ...
3
votes
0answers
27 views

Difference between almost everywhere convergence of whole Fourier series and a subseries of $L^2$ functions

Let $f \in L^2(\mathbb{R})$ and $F_{N}f$ denote the $N$-th partial sum of its Fourier series. Then $||F_{N}f -f||_{L^{2}} \rightarrow 0$ as $N \rightarrow \infty$. But this implies there exists a ...
0
votes
0answers
39 views

Homework excercise, completeness in Vector-spaces, is it correct?, long, but can it be simplified?

I have a very difficult excercise. I see now that it became too much text for someone to might go through it, if you can please help me, but don't want to read all, can you please then only answer my ...
0
votes
0answers
39 views

Show convergence of an infinite sum by ratio test

I would like to show that $$\sum\limits_{n=1}^\infty \frac{1}{|n^x|}$$ converges. I was hoping to do this by the ratio test, since we haven't covered the integral test yet so we are not allowed to use ...
1
vote
0answers
24 views

Comparison of the remainder of a series and its general term

Let $\sum\limits_{n = 0}^{+ \infty} u_{n}$ be a convergent series, such that $\forall n , u_{n} > 0$ My question is : Under which conditions can we find a constant $C > 0$ such that $$\forall ...
-1
votes
2answers
56 views

How to generate a sequence that converges to $1/n$ where $n \in \mathbb{N}$

I am curious how to generate a sequence $(a_n)$ that converges to $1/k$ for any $k \in \mathbb{N}$ when $n \to \infty$. Of course one can think of a sequence that converges to $1/k$ for specific $k$, ...