Tagged Questions
0
votes
1answer
32 views
Show convergence for this sequence only by using the definition
I need to prove convergence for
$(b_n)_{n ∈ ℕ}=\left(\frac{(-1)^nn}{2n+1}\right)_{n∈ℕ}$ and also show the limit.
I may only use the following definition: $∀ɛ > 0∃n_0∈ℕ∀n≥n_0:|a_n-a|< ɛ$.
So far ...
3
votes
1answer
27 views
Evaluating order of convergence
I think this is quite a simple question, I just want to make sure I understood all correctly.
Here's the problem: I have a numerical method, which is in some way dependent on its spacing $h$ (like ...
2
votes
2answers
83 views
Using the Banach Fixed Point Theorem to prove convergence of a sequence
Use the Banach fixed point theorem to show that
the following sequence converges. What is the limit of this
sequence?
$$\left(\frac{1}{3},
\frac{1}{3+\frac{1}{3}},
...
0
votes
1answer
48 views
How to prove a telescoping series converges ???
Let ${a_k}$ be a real sequence, such that $a_k \rightarrow 3, $as k$ $$\rightarrow \infty$ .
Prove that $\sum_{k=1}^\infty (a_k- a_{k+3}) = a_1 + a_2 +a_3 - 9$.
I know this is a telescoping series ...
0
votes
1answer
36 views
Convergence of sum of random variables
Let $X_n$, $n\geq 0$, be i.i.d. random variables such that: $\mathbb E(X_1)=0$, and $0<\mathbb E(|X_1|^2)<\infty$. Given that $\alpha >\frac{1}{2}$, I need to show that ...
1
vote
2answers
30 views
Convergence of infinite series with p-test and constant
My question is: Does the infinite series
$\sum_{n=1}^\infty \frac{1}{n^{\frac{4}{5}}+10^{10}}$
converge or diverge?
I know that $\frac{1}{n^{\frac{4}{5}}}$ diverges by the $p$-test, and that adding ...
2
votes
2answers
69 views
Real analysis - converging sequence [duplicate]
My answer
Solution
1).
$Let\; \epsilon = L/2 > 0 \mbox{thus by definition of}\; x_m→L, \mbox{there exists}\; a \;n_o∈ N \;\mbox{such that }∀m>n_o\; \\
|x_m - L|< ε\\
-ε <|x_m - L| < ...
-1
votes
1answer
42 views
Convergence in L 1
Prove that if the $X_l$'s are i.i.d. and in $L^1$, then $(n^{-1} \sum^n_{k=1} X_k)_{n \ge 1}$ is uniformly integrable .
gb
2
votes
3answers
97 views
How to solve $\sum_{k=2}^\infty {\frac{1}{k^2-1}}$
I'm using the integral test to determine if this series converges. From what I have so far it seems that it diverges, but according to wolfram alpha it converges. Where is my mistake?
...
1
vote
2answers
46 views
Convergent sequence question…
I have a homework question that I am not sure how to begin. We are asked to suppose $\{a_n\}$ and $\{b_n\}$ are sequences such that $\{a_n^2 + b_n^2\} \rightarrow 0$.
We have to prove $\{a_n\} ...
1
vote
2answers
38 views
If $x\in X$ and sequence $(x_n) \in X^{\Bbb N}$ converges in $(X,d)$ , then so does every subsequence of $(x_n)$.
A subequence of a sequence $(x_n)_{n\ge 1}$ is a sequence $ (x_{n_1}, x_{n_2},x_{n_3},...$) where $n_1,n_2,n_3,... \in \Bbb N$ with $n_1\lt n_2\lt n_3\lt ...$
Let $(X,d)$ be a metric space and let ...
3
votes
2answers
71 views
Kolmogorov's maximal inequality and convergence of random series.
Let $(X_n)_{n\ge 1}$ be a sequence of mutually independent random variables, on the same probability space, with expectation 0 and finite variance. Let $S_n = \sum_{l=1}^n X_l$. Prove that for any ...
1
vote
1answer
59 views
Point convergence, uniform convergence and near uniform convergence of infinite series $ f_n = x^2 e^{-nx}$
Please help me in prove / decline the point convergence, uniform convergence and near uniform convergence (comapact uniform convergence) of
$\sum_{n=1}^{\infty} f_n$ where
$f_n : [0, +\infty) ...
1
vote
2answers
31 views
Define $f: I \rightarrow \mathbb{R}$ as $f(x)= \sup {f_n(x) : n \geq n_0 }$ for $ x \in I$, It's convex?
Suppose, that $f_n:I\rightarrow \mathbb{R}$ are convex functions for $n\geq n_0$ and
$\forall_{x\in I} \exists_{y\in \mathbb{R}} \forall_{n\geq n_0} f_n(x)\leq y$
Define $f: I \rightarrow ...
5
votes
3answers
72 views
$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n \cdot g_n$ uniformly converge to $f\cdot g$
Please help me check, if
$f_n$ uniformly converge to $f$ and $g_n$ uniformly converge to $g$ then $f_n + g_n$ uniformly converge to $f+g$
$f_n$ uniformly converge to $f$ and $g_n$ uniformly ...
0
votes
1answer
63 views
Convergent & Cauchy Sequence related prove
(1) Consider the two convergent sequences $\{a_n\}$and $\{b_n\}$ such that $\{a_n\}\to a$ and $\{b_n\}\to b$.
Prove that $\{a_n+b_n\}\to a + b$
(2) Prove that a convergent sequence is Cauchy. ...
3
votes
0answers
39 views
How to show $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
How can I construct a random variable $X$ such that: $nP\{|X|>n\}\to 0$ as $n\to\infty$, but $X$ is not integrable.
3
votes
2answers
52 views
Let $\{X_n\}$ be i.i.d integrable r.v.s, show that $\frac{1}{n}\max_{1\leq j\leq n}|X_j|\to 0 \quad \mbox{a.e.}$
This problem is an exercise in Probability theory,independence,interchangeable, martingale(Chow), exercise 4.1.10.
Let $\{X_n,n\geq 1\}$ be independent identical distributed integrable random ...
2
votes
1answer
38 views
Convergent series and of positive integers and partial sums.
Let $\sum a_n$ be a convergent series of positive real numbers with sum $s$ and partial sums $s_n=a_1+a_2+\cdots+a_n$.
Prove that $\sum na_n$ is convergent if and only if $\sum (s-s_n)$ is ...
2
votes
3answers
91 views
Does this series converge or diverge?
I have a series here, and I'm supposed to determine whether it converges or diverges. I've tried the different tests, but I can't quite get the answer.
...
2
votes
3answers
69 views
Do the sequences from the ratio and root tests converge to the same limit?
For example, if we have:
$$\sum_{n=1}^{\infty}a_n$$
Where the ratio test is satisfied. That is $\exists L$ s.t.
$$\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = L < 1$$
Does this ...
4
votes
1answer
108 views
Need to prove the sequence $a_n=(1+\frac{1}{n})^n$ converges by proving it is a Cauchy sequence
I am trying to prove that the sequence $a_n=(1+\frac{1}{n})^n$ converges by proving that it is a Cauchy sequence.
I don't get very far, see: for $\epsilon>0$ there must exist $N$ such that ...
2
votes
2answers
94 views
Kernel of $p$-adic logarithm.
I'm completely clueless as to how to answer the following question:
Let $K$ be a field of characteristic zero which is complete with respect to a non-Archimedean aboslute value $|\cdot|$. Let ...
0
votes
1answer
25 views
Weak and normwise convergence of sequence of linear functionals
Is this sequence of linear functionals weakly (normwise) convergent : $$f_n((x_j))=\sum_{k=1}^{n}{\frac{x_k}{k}} , (x_j) \in \ell_1\,?$$
5
votes
3answers
64 views
Representing Functions as Power Series
Rewrite $$f(x)=(1+x)/(1-x)^2$$ as a power series.
Work thus far:
I separated it into two parts:
$$1/(1-x)^2 + x/(1-x)^2$$
I realize that the first expression is the derivative of $1/(1-x)$ and ...
1
vote
2answers
81 views
Show that $f(x)=\sum_{k=1}^\infty \frac{1}{k}\sin(\frac{x}{k+1})$ converges.
Exercise: Show that
$$f(x)=\sum_{k=1}^\infty \frac{1}{k}\sin\left(\frac{x}{k+1}\right)$$
converges, pointwise on $\mathbb{R}$ and uniformly on each bounded interval in $\mathbb{R}$, to a ...
0
votes
1answer
39 views
Sequence with integral values ultimately constant
Dear Stackexchange Community,
I would like to ask how to show that a Cauchy sequence that takes on integral values is ultimately constant.
Being unfamiliar with this, do 'integral' values refer to ...
1
vote
1answer
59 views
Regarding an isomorphism between a subspace of $\ell^{\infty}$ and $\ell^1$
Let $c_0$ be the subspace of $\ell^\infty$ consisting of sequences that converge to $0$. Show that $c_0$ is a closed subspace of $\ell^{\infty}$ whose dual space is isomorphic to $\ell^1$. Conclude ...
0
votes
1answer
51 views
Conditions that imply Lindeberg's condition
Suppose that $Z_{1},Z_{2},\ldots$
are i.i.d. random variables with mean 0 and variance 1, and define $X_{nk}=\sigma_{nk}\cdot Z_{k}$
If
$$
\frac{\underset{1\leq k\leq ...
1
vote
1answer
62 views
Weak Convergence to Exponential Random Variable
Assume that $X_1$, $X_2$,... are independent random variables uniformly distributed on $[0,1]$. Let $Y^{(n)}=n\inf\{X_i,1\leq i\leq n\}$. I am asked to prove that it converges weakly to an exponential ...
1
vote
1answer
50 views
Sequential continuity in normed linear spaces
I am trying to prove the following "contiuity-type" result.
Let $X,Y$ normed linear spaces. Let $\{T_n\} \to T \in \mathcal{L}(X,Y)$ and $\{u_n\} \to u \in X$. Show that $\{T_n(u_n)\} \to \{T(u)\} ...
1
vote
1answer
63 views
Sequences in Banach spaces
I am very bad with proofs that ask you to show that "$\exists$ something..." because in most of them you have to explicitly show the something. The following question is one such. Any help will be ...
3
votes
1answer
54 views
Hausdorff iff convergent filterbase has a unique point of convergence. Is this correct?
Suppose $(X, \mathcal{T})$ is a Hausdorff space. Consider a filterbase $\mathcal{B}$ in $X$ such that $\mathcal{B}$ converges to $a\in X$.We show that, if $b \in X$ and $a\neq b$, then $\mathcal{B}$ ...
2
votes
1answer
106 views
applying multi-section formula to find convergence
The question asks to use the multi-section technique to determine if
$$\sum_{n>=0} (a^n)/(4n +1)!$$
converges, and to provide a finite expression for the exact value of the series.
The multi ...
0
votes
1answer
51 views
Failure of convergence to 0
Consider the interval $I = [0,1]$ and the sequence of functions:
$$f_n(x) = (-1)^k \ \text{for} \displaystyle \frac{k}{2^n} \le x < \frac{k+1}{2^n} \ \text{where} \ 0 \le k < 2^n - 1$$
I want ...
3
votes
2answers
146 views
Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$,$\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ converge, their product converges too
Show if $\displaystyle\sum\limits_{k=1}^\infty {a_k}^2$ and $\displaystyle\sum\limits_{k=1}^\infty {b_k}^2$ both converge, $\displaystyle\sum\limits_{k=1}^\infty {a_kb_k}$ also converge
then
Show if ...
0
votes
1answer
46 views
Prove that $(X_1 X_2\cdots X_n)^{1/n} \to c$ as $n\to\infty$ where $c$ is a constant
This is a assignment question, a part of my homework. So I need hints to start towards the solution. I was thinking that under the given conditions of the problem the random variables $\log X_1$, ...
3
votes
1answer
72 views
Questions regarding the sequence $f_n(x)=\frac{x^2}{x^2+(1-nx)^2},\ \ \ \ \ x\in[0,1]$
Define a function $f_n:[0,1]\to \mathbb{R}$ by
$$f_n(x)=\frac{x^2}{x^2+(1-nx)^2},\ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in[0,1]$$
I am asked three questions:
(a) show that the sequence $(f_n)$ is ...
1
vote
3answers
87 views
Uniform convergence of a sequence of functions
Let $(u_n(x))$ be a sequence of functions in $(0,\infty)$ such that:
$u_1(x)=x$, $u_{n+1}=\frac12\left(u_n(x)+\frac1{u_n(x)}\right)$ for $n\in\mathbb{N}$.
Check if $u_n(x)$ converge unfiormly in ...
2
votes
2answers
50 views
Convergence of an Ergodic process
I'm having trouble working through the math of the problem below. I believe the problem as described is an ergodic process. I've written a simple simulation of the problem, that converges to 66.6...6% ...
0
votes
1answer
95 views
Function Series Pointwise and Uniform convergence
Check the function series $(f_{n})_{n \in \mathbb{N}}, f_{n}:]0,\infty[ \rightarrow \mathbb{R}$ for pointwise and uniform convergence and prove your results.
a) $f_{ n }\left( x \right) =\frac { n^{ ...
3
votes
1answer
177 views
Power Series and Radius of Convergence
Determine for following Power Series in $\mathbb{C}$ the radius of
Convergence.
a) $\sum _{ n=0 }^{ \infty }{ (2+\sqrt { n } )^{ n }z^{ n } } $
b) $\sum _{ n=1 }^{ \infty }{ (1-\frac ...
0
votes
1answer
38 views
Show that $P(X_n \leq x_n) \rightarrow P(X\leq x)$
Suppose $X_n$ converges in distribution to $X$, $x_n\rightarrow x$ and the cumulative distribution function for $X$ is continuous at $x$. Show that $P(X_n \leq x_n) \rightarrow P(X\leq x)$
2
votes
1answer
61 views
Showing convergence in Space of Squared Summable Sequences
My Problem: Show that the sequence ${x_n}_{n\geq 1}$, where $x_n=(1,\frac{1}{2},\ldots,\frac{1}{n},0,0,\ldots)$ converges to $x=(1,\frac{1}{2},\frac{1}{3},\ldots,\frac{1}{n},\ldots)$ in $l_2$
My ...
0
votes
4answers
84 views
Another series convergence question
Does this series converge?
$\displaystyle\sum\limits_{n=2}^\infty \frac{\sqrt{n+1}}{(2n^2-3n+1) (\ln n +(\ln n)^2)}$
4
votes
4answers
88 views
Another simple series convergence question
I'm being asked to determine if $\displaystyle\sum\limits_{n=3}^\infty \frac1{n (\ln n)\ln(\ln n)}$ converges. So, using Cauchy's Condensation Test, I reduced the problem to one of determining the ...
2
votes
2answers
77 views
A sequence in $C([-1,1])$ and $C^1([-1,1])$ with star-weak convergence w.r.t. to one space, but not the other
The functionals
$$
\phi_n(x) = \int_{\frac{1}{n} \le |t| \le 1} \frac{x(t)}{t} \mathrm{d} t
$$
define a sequence of functionls in $C([-1,1])$ and $C^1([-1,1])$.
a) Show that $(\phi_n)$ converges ...
1
vote
0answers
22 views
Convergence of a serie depending on an exponent
For which $k>0$ is $\sum \limits_{n=1}^\infty (\frac {1}{n+n^k})$ convergent.
I tried to prove convergence with the Ration test but it hasn't really worked.
by the P-test and comparison test I ...
0
votes
1answer
28 views
Prove of convergence of a serie
Prove if that serie is convergence:
$\sum \limits_{k=1}^\infty (\frac{k^3}{-1-k^4})$
$<=>\sum \limits_{k=1}^\infty -(\frac{k^3}{k^4+1})$
My problem is that I don't know how to do that I tried ...
0
votes
2answers
63 views
Convergence of series 1
Determine wether these series are convergent:
1) $$\sum_{l=5}^\infty \left(\frac{1}{l^2} + \frac {2}{l^3}\right)$$
$$ \left(\frac{1}{l^2} + \frac {2}{l^3}\right)=\frac {l^3+l^2}{l^5} > \frac ...
