Convergence of sequences and different modes of convergence.

learn more… | top users | synonyms

6
votes
1answer
73 views

Prove series converge for almost every $x$

Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum_{n=1}^{\infty}\int_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in ...
0
votes
3answers
44 views

Show pointwise convergence and (potentially) uniform convergence $\sum_{k=1}^\infty\frac{x^k}{k}$

I am looking to show pointwise convergence and (potentially) uniform convergence of the following: $$\sum_{k=1}^\infty\frac{x^k}{k}$$ I know (from my book) this converges for my given values of $x ...
0
votes
0answers
26 views

Find where {fn(x)} n=1 to infinity converges pointwise

fn(x) = $ \frac{x}{(1+x)}$ and x ∈ R Find all real numbers x where fn(x) converges pointwise and describe the limit function. Any hints or suggestions?
2
votes
2answers
17 views

$X_n\to 0$ in probability implies $E[f(X_n)]\to f(0)$ for $f$ uniformly cts and bounded

Let $f$ be a uniformly continuous and bounded function. I've shown that if $X_n\to 0$ in probability, then $f(X_n)\to f(0)$ in probability as well. Now I want to say that $$\lim_{n\to\infty} ...
0
votes
1answer
62 views

Does a convergent sequence in theory ever reach its limit?

Completing a question on the sequence $\{a_n\} = \frac n{2n+1}$. Does $a_n$ in this sequence ever actually get to $\frac12$ officially?
1
vote
1answer
19 views

Sequences of random variables converging in probability to the same limit a.s.

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, ...
2
votes
1answer
30 views

Uniform convergence on compact subset

Let there be two functional squences $$a_n(x)=\sqrt[n]{x} \quad \textrm{ for $x\in(0,\infty)$}$$ $$b_n(x)=\sum_{k=0}^{n}x^k(1-x)^k=\frac{1-x^{n+1}(1-x)^{n+1}}{x^2-x+1} \quad \textrm{ for $x\in ...
2
votes
1answer
33 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
0
votes
1answer
29 views

Uniform and pointwise convergence

So we had the pointwise and uniform convergence, and I do get that a sequence of function can converge to a function, just like ordinary sequences do. But what I don't quite get is this pointwise and ...
0
votes
1answer
49 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
0
votes
1answer
27 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
-1
votes
0answers
49 views

$X_n$ doesn't converge to a limit in $[-\infty, \infty] \to$ Is this supposed to be a stronger version of $\lim X_n$ doesn't exist?

From Williams' Probability with Martingales: What's the difference between saying that '$X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$' and '$\lim X_n$ does not exist' ? ...
0
votes
3answers
53 views

Finding a Taylor Series Representation: $f(x)= \frac{x}{(1+4x^2)^2}$ [closed]

Find Taylor series representations for the following function. For precisely what values of $x$ is the series representation valid? $$f(x) = \frac{x}{(1+4x^2)^2}$$
3
votes
1answer
34 views

Definition of convergence of a sequence

Can this be a valid definition? For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$. Should I use "$<\epsilon$" or "$\le \epsilon$"?
3
votes
3answers
33 views

A different notion of convergence for this sequence?

I was thinking about sequences, and my mind came to one defined like this: -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, ... Where the first term is -1, and after the nth occurrence of -1 in the ...
-1
votes
2answers
49 views

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? [duplicate]

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? This is what I have done, since $0\le x\le 1$ $ x=0$$$f_n(0)=0-0^n=0$$ $x\to \infty$ ...
1
vote
0answers
29 views

Convergence of $\sum \frac{a_n}{1+a_n}$ implies convergence of $\sum a_n$ for positive $a_n$. [duplicate]

I need to prove or disprove the statement. I think the statement is true. My attempt at a proof: From the definition of convergence: $$\forall \epsilon > 0 \quad \exists N \in \mathbb{N} \quad ...
0
votes
0answers
16 views

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$ converges if the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \rvert$ converges too and the ...
2
votes
3answers
87 views

Does the infinite series $\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$ converge?

I have been wondering if this infinite series converges $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$$ I tried to put it in wolfram alpha but it says that the ratio test is inconclusive, but when I ...
0
votes
1answer
25 views

Should monotone convergence theorem say uniformly bounded?

saz pointed out to me the difference between bounded and uniformly bounded: $Y$ is uniformly bounded: there exists $C>0$ such that $|Y_n| \leq C$ for all $n \in \mathbb{N}$, i.e. $$|Y_n| ...
0
votes
0answers
20 views

sum converge, matrix, norm

Let $A_j$ be a sequence in $\mathbb{C}^{n\times n}$. Show that $ \sum_{j=0}^\infty A_j$ converges if $ \sum_{j=0}^\infty ||A_j||$ does.($||A||= sup_{|x|=1} |Ax|$ with euclid norm) Hello, Be ...
0
votes
1answer
20 views

Question Concerning Fourier Series

I was following the derivation of the basic Fourier series using orthogonal function. For the set of orthogonal functions $\{\phi_n\}$, say the function $f$ can be defined as: $$f(x) = c_0 \phi_0(x) ...
1
vote
2answers
50 views

Convergent subsequence in a bounded sequence of a complete metric space

Consider a complete metric space E with the following property: If $x_n$ is a bounded sequence, then $\forall \epsilon > 0$, $\exists i,j , i \neq j$ such that $d(x_i,x_j) < \epsilon$. ...
1
vote
2answers
39 views

Sum of reciprocals of prime-index-primes

Let $p_1=2$, $p_2=3$, $p_3=5$, $\ldots$ be an enumeration the prime numbers. If $q$ is a prime number, we call $p_q$ a prime-index-prime. A list of prime-index-primes can be found here. My question ...
2
votes
1answer
34 views

If $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} a_{n}$ converges.

Let $a_{n} \geq 0$ and $b_{n}>0$ for each $n$ in $\mathbb{N}$ and suppose that $\lim_{n \to \infty} \frac{a_{n}}{b_{n}} = 0$ and $\sum_{n=1}^{\infty} b_{n}$ converges, then $\sum_{n=1}^{\infty} ...
0
votes
0answers
24 views

Convergence of a function of two variables

The following question has been posed to me by a student in an analysis class. For which real numbers $\alpha \gt 0$ is the function $f : \Bbb R^2 \to \Bbb R$ given by $f(x, y) = (x^2 + y^2)^\alpha$ ...
4
votes
2answers
59 views

Areas under the graphs of $\frac{1}{x}$ and $\frac{1}{x^2}$ from $1$ to $\infty$

A simple evaluation of the definite integral tells us that the area under the graph of $[\frac{1}{x}]^2$ from $1$ to $\infty$ is finite whereas that of $\frac{1}{x}$ for the same limits is infinite. ...
1
vote
2answers
34 views

Convergence from another series

Suppose that both $$\sum_{j=0}^n a_j^2$$ and $$\sum_{j=0}^n b_j^2$$ are convergent. Show that $$\sum_{j=0}^n a_jb_j$$ converges absolutely. Ok, so my final exam is tomorrow and I have been working on ...
1
vote
3answers
57 views

Where does the following series converge? [closed]

Using integrals or by any other method find: $\lim_{n \rightarrow\infty} \sum_{i=1}^{n}\frac{1}{n+i}$
0
votes
1answer
24 views

Convergence of $f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1]$

$(f_n)$ is a succession of functions $$ f_n : [-1,1] \rightarrow \mathbb{R} \\ \\ f_n (x)=\sqrt[3]{\rvert{x}\rvert ^3+\frac{1}{n}} \qquad x \in [-1,1] $$ Punctual convergence $\forall x \in ...
7
votes
0answers
81 views

Which functions are irrelevant?

Call a bijection $f : \mathbb{N} \rightarrow \mathbb{N}$ irrelevant over $\mathbb{R}$ iff for all sequences $a : \mathbb{N} \rightarrow \mathbb{R}$, if $$\sum_{i=0}^\infty a_n$$ exists, call its value ...
0
votes
0answers
13 views

Limit laws for power of convergent sequence

Let $ \{ a_n \}_{n=m}^{\infty}$ be convergent sequence of real numbers such that $\lim _{n \rightarrow \infty}a_{n}=x$. Can we say that $\lim _{n \rightarrow \infty}a_{n}^q=(\lim _{n \rightarrow ...
0
votes
0answers
22 views

Newton Method Variant with convergence of order 3

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be twice continuously differentiable for all $x$ in the neighborhood $U(\xi)=\{x\in\mathbb{R}:|x-\xi|<r\}$ of a simple zero $\xi$ of $f$ such that ...
0
votes
2answers
47 views

For what values of $k$ to both of the following series converge?

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
2
votes
0answers
94 views

For which $s$ is defined $\frac{1}{\zeta(s)}\sum_{n=1}^\infty\frac{n}{ \left( \zeta(s) \right) ^n}$, where $\zeta(s)$ is the Riemann Zeta function?

If there are no mistakes in my words, if one of the factors $\sum_{n=1}^\infty a_n$ of a convolution product or Cauchy product converges, and the other $\sum_{n=1}^\infty b_n$ corverges absolutely ...
-1
votes
1answer
28 views

Sequence converging to a closed set

How can I prove that A sequence generated by a particular algorithm converges to a set? i.e irrespective of the starting point the sequence converges to any of the points in a set, or oscillates ...
1
vote
5answers
56 views

How to prove that $\lim\limits_{n\to\infty}\sqrt[n]{p}=1$? [closed]

Could you tell me how to show if $p>0$ then$\lim\limits_{n\to\infty}\sqrt[n]{p}=1$? (+clues) 1.put $\sqrt[n]{p}=1+h_{n}$ 2.Bernoulli's inequality If you don't mind, use the clues to prove it.
0
votes
0answers
39 views

Determine whether the following sequences (fn) converge uniformly, pointwise, or neither:

Determine whether the following sequences $(f_n) \in F(E, \mathbb{R})$ - where E is a set - converge uniformly, pointwise, or neither: a) $f_n(x) = \frac{n^2x} { 1 + n^2x^2}$ on set $E = \mathbb{R}$ ...
0
votes
1answer
45 views

Radius of Convergence of $\sum_{n = 0}^{\infty} \frac{(-1)^nn!x^n}{n^n}$

I'm taking the AP Calculus BC Exam next week and ran into this problem with no idea how to solve it. Unfortunately, the answer key didn't provide explanations, and I'd really, really appreciate it if ...
0
votes
2answers
45 views

Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$

Investigate convergence of $\sum_{n=1}^\infty \frac{\ln(n)}{n}$ I applied nth term test and was inconclusive. I tried ratio test but I don't know how to evaluate the limit. I think it is 1 therefore ...
0
votes
1answer
38 views

Investigate convergence of $\sum_{n=2}^\infty \frac{1}{n(n-1)}$

Investigate convergence of: $\sum_{n=2}^\infty \frac{1}{n(n-1)}$ Now I know by $p$-series test this summation converges however, is there a way to prove that this series converges by some ...
2
votes
0answers
11 views

When does the cohomological Atiyah–Hirzebruch–Leray–Serre spectral sequence converge?

Given a Serre fibration $F \to E \to B$ of spaces homotopy equivalent to CW complexes, with $B$ simply-connected, and a generalized homology theory $h_*$ with respect to which the fibration is ...
5
votes
0answers
74 views

Can the boy escape the teacher for a regular $n$-gon?

This is related to Prove that the boy cannot escape the teacher Suppose there is a boy in the center of a regular $n$-gon. The teacher is on the edge of the $n$-gon (but cannot leave the edge) and ...
3
votes
2answers
51 views

Determine wether $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ converges or diverges.

Determine wether the following function converges or diverges by comparison test: $\sum_{n=1}^\infty \frac{3n-1}{(n+1)^3}$ Upon inspection I can clearly see that the series converges. However I am ...
2
votes
1answer
34 views

Proof of a theorem regarding subsequences and convergence

I am attempting to understand the following theorem and it's proof as outlined in my textbook for Real Analysis. Theorem: Let $(s_n)$ be a sequence. If t is in $\mathbb{R}$, then there is a ...
6
votes
1answer
42 views

$\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e.

Let $\mu(X) \lt \infty$. Then $f_k \to f$ in measure iff for any subsequence $k_l$, there is a subsequence $k_{l_n}$ such that $f_{k_{l_n}}\to f$ a.e. I can show the only if part by using the theorem ...
1
vote
1answer
26 views

Let $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ in measure.

Let $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ in measure. I looked at the proof of this statement and it says that it follows from the fact that if $f_k \to f$ in $L^{\infty}$, then $f_k \to f$ ...
1
vote
1answer
18 views

Give necessary and sufficient conditions so the sum of random variables converges almost surely

$\{X_k\}_{k}$ are independent random variables on the probability space $(\Omega, \mathcal F, P)$ and $X_k$ has gamma density $f_k(x)$ where $f_k(x)=\dfrac{x^{a_x-1}e^{-x}}{\Gamma(a_k)}$ where $x,a_k ...
1
vote
1answer
29 views

Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$

Find the radius of convergence of $\sum_{n=1}^\infty n!(2x-1)^n$ Now, by D'Alemberts Ratio test that implies (for convergence): $\lim_{n\to\infty} \lvert \frac{(n+1)!(2x-1)(2x-1)^n}{n!(2x-1)^n} ...
1
vote
1answer
31 views

Find convergence domain of the integral

Find convergence domain of $$\int_0^\infty \! \frac{\cos^2{x}}{x^p} \, \mathrm{d}x$$ I've tried to use $\frac{\cos^2{x}}{x^p} < \frac{1}{x^p}$, but $\int_0^\infty \! \frac{1}{x^p} \, \mathrm{d}x$ ...