Convergence of sequences and different modes of convergence.
0
votes
1answer
33 views
Order of convergence of a sum
Let $(X_t)_{t\geq 0},\;X_0=0$, be a positive stochastic process such that
\begin{align*}
\mathbb{E}\left[\sum_{n=1}^{\infty}X_t^n\right]=\sum_{n=1}^{\infty}\mathbb{E}[X_t^n]<\infty.
\end{align*}
...
0
votes
0answers
27 views
$C_{c}(X)$ is a subspace of $C_{0}(X)$ but it is not Banach [duplicate]
I want to show that $C_{c}(X)$ is a subspace of $C_{0}(X)$ but it is not Banach.
I think : I should prove that if $\{f_{n}\}$ be a sequence in $C_{c}(X)$ with ...
1
vote
1answer
22 views
Example of sequence converging in $d_{l^\infty}$ but not in $d_{l^1}$.
I'll denote by $X$ the space of real sequences $(a_n)$ such that $\sum |a_n|$ converges. Let $d_{l^1}$ be the metric
$$
d_{l^1}((a_n),(b_n))=\sum|a_n-b_n|
$$
and $d_{l^\infty}$ be the metric
$$
...
0
votes
0answers
29 views
Are these series convergent?
I came across the following two series while trying to solve Laplace's equation in two dimensions.
$$T_1 = \sum_{n=1}^{\infty}\rho^n(A_n\cos(n\phi) + B_n \sin(n\phi))$$
$$T_2 = ...
1
vote
0answers
28 views
measure theory and convergence
1) Let $\Omega=[0,1]$, $F = B([0,1])$, $P$ be Lebesgue measure on $[0,1]$ ($P([a,b])=b-a$).
Set
$$A_n^i:=\left[{\frac{i-1}{n},\frac{i}{n}}\right]$$
and
$$X_n^i(\omega):=\chi_{A_n^i}(\omega)$$
...
2
votes
2answers
37 views
Find an interval of convergence and an explicit formula for $f(x)$
Let $f(x) = 1 + 2x + x^2 + 2x^3 +x^4+...$
If $c_{2n} = 1$ and $c_{2n+1} = 2$ $\forall n \ge 0$ find the interval of convergence and an explicit formula for $f(x)$.
The answers are $I = (-1,1)$ and ...
5
votes
1answer
46 views
$\int_\Omega F(u_n)\to0$ implies $\int_\Omega F(au_n)\to 0$ for $a\in [0,\infty)$?
Let $\Omega\subset\mathbb{R}^N$ be a bounded domain and $F:[0,\infty)\to [0,\infty)$ be a convex, strictly increasing and continuous function satisfying $F(0)=0$. Suppose that $u_n\in ...
0
votes
0answers
29 views
Solving $m$ in $m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}}$ from $n$ and $x$
How should one proceed in order to solve $m$, where $x$ is an integer
$$m = \lim_{n\to\infty}\prod_{k=x+1}^n\, 1+\dfrac{(k+x)^2}{2^{k-x}} $$
from $n$ and $x$ in an unconditional form, such as, for ...
3
votes
2answers
71 views
Definition of convergence in $C^\infty(\Omega)$
I am not convinced or may be don't understand, the way they define convergence and then topology as a consequence of convergence.
$\Omega$ is open subset of $\Bbb R^n.$Define standard topology on ...
7
votes
1answer
101 views
convergence of series with $k!$
check if the following series converges:
$\sum\limits_{k=1}^{\infty} (-1)^k \dfrac{(k-1)!!}{k!!}$
where $k!!=k(k-2)(k-4)(k-6)...$
I came across this exercise while going trough some old exams. I'm ...
1
vote
1answer
29 views
Show that Y is a closed subspace of l2
This might be a straight forward problem but I wouldn't ask if I knew how to continue.
Apologies in advance, I am not sure how to use the mathematical formatting.
We are currently busy with inner ...
2
votes
3answers
101 views
Convergence of $\sum \frac{a_n}{(a_1+\ldots+a_n)^2}$
Assume that $0 < a_n \leq 1$ and that $\sum a_n=\infty$. Is it true that
$$ \sum_{n \geq 1} \frac{a_n}{(a_1+\ldots+a_n)^2} < \infty $$
?
I think it is but I can't prove it. Of course if $a_n ...
2
votes
1answer
31 views
Intuition behind closed subsets of a metric space?
Reading for my exam in real analysis, I struggle with the definition of a closed subset of a metric space.
Consider a metric space $$(X,d)$$
Then consider a subset of this space$$F$$
What the book ...
0
votes
0answers
40 views
funcitonal series convergence… SOS… [duplicate]
Is it possible to show that $f(x) = \sum_{k=0}^\infty (-1)^k \frac{x^k \sqrt{k}}{k!}$ converges even when $x\rightarrow \infty$ ?
i.e. As a function of x, does $\lim_{x\to\infty} f(x)$ exist or at ...
-1
votes
1answer
38 views
Alternating functional Series Convergence SOS…
Does the following series converge?
$\sum_{k=0}^\infty \frac{(-1)^k x^k \sqrt{k}}{k!}$
what is the radius of convergence?!!
0
votes
1answer
41 views
Multiplication in $\mathcal D'(R)$.
I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
1
vote
1answer
46 views
Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.
I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
1
vote
0answers
13 views
Convergence in distribution of a random variable/vector
I'm trying to prove a version of Slutsky theorem and I appreciate if you can guide me on that.
Suppose $X_n \rightarrow^{d} X > 0$ (i.e. $X_n$ convergence in distribution to X which is positive) ...
0
votes
1answer
62 views
Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$
Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this ...
1
vote
4answers
39 views
Is $(a_n)_n$ with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ a Cauchy sequence?
Let $0 < q < 1$ and $(a_n)_n$ be a squence with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ for all $n ∈ ℕ$.
I need to show that this is a Cauchy sequence. I'm not sure how to start this one, as we ...
0
votes
1answer
33 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 ...
0
votes
3answers
38 views
Evaluating $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$
I am trying to evaluate $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$. Factoring and using partial fraction decomposition, I have found that the indefinite integral is:
$$\ln{|x-5|}-\ln{|x-3|} + C$$
But ...
1
vote
2answers
46 views
Prove the convergence of the sequence.
Prove the convergence of the following sequence:
$$x_1 = \sqrt{a}$$
$$x_{n+1} = \sqrt{a + x_n}$$
3
votes
1answer
29 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 ...
5
votes
4answers
112 views
Evaluating $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$
Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$?
I've calculated that the recurrence relation for this integral is:
$\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} ...
1
vote
1answer
29 views
series convergence
i ran into this question:
prove or show false:
if $\sum_{n=1}^{\infty}a_{n}$ is a converging series, but the series $\sum_{n=1}^{\infty}a_{n}^2$ diverges, then $\sum_{n=1}^{\infty}a_{n}$ is ...
7
votes
1answer
74 views
methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$
As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which ...
2
votes
0answers
9 views
Polynomials, integrals convergence
Let $P_n(x)= \frac{x^n(bx -a)^n}{n!}, \ \ \ a,b,n \in \mathbb{N}$.
Prove that $\int_0 ^{\pi}P_n(x) \sin xdx \rightarrow 0 \ \ \ \ $ and $ \ \ \ \ \int_0 ^r P_n(x)e^xdx \rightarrow 0$ $ \ \ \ \ \ \ (n ...
1
vote
1answer
30 views
A problem on almost sure convergence
Consider a sequence of random variables defined on the standard unit interval probability space :
$ X_n = 2^n \text{when} \frac{1}{2^n} \leq \omega \leq \frac{1}{2^{n-1}}$
...
0
votes
2answers
43 views
Checking $\displaystyle \int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$ for convergence
Given $\displaystyle\int_{0}^{\infty} {\frac{\sin^2x}{\sqrt[3]{x^7 + 1}} dx}$, prove that it converges.
So of course, I said:
We have to calculate $\displaystyle \lim_{b \to \infty} ...
0
votes
2answers
20 views
Develop the next function:$f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0)$
Develop the next function:$\displaystyle f(x)=\frac{4x+53}{x^2-x-30}$ into power series, Find the radius on convergence and find $f^{(20)}(0).$
For the first part:
$\displaystyle\frac ...
1
vote
2answers
62 views
How to show that these integrals converge?
What test do I use to show that the following integral converges?
If you could provide me with the process that leads to the answer that would really help.
$\displaystyle ...
5
votes
2answers
48 views
“Nearly” Harmonic Series
It's well known that
$$
\sum_{n=1}^{\infty} \frac{1}{n^{1+\varepsilon}} < \infty, \ \forall \varepsilon >0.
$$
What happens if we replace $\varepsilon$ with $\varepsilon_n \downarrow 0$?
...
2
votes
1answer
34 views
Convergence of sequence
Does the following:
$$
\begin{align}
x_0 & = a \\
x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\
x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\
x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) ...
0
votes
1answer
26 views
Convergence of random variable to a negative constant
Let $X_n$ be the sequence of R.Vs and $X_n\overset{P}{\rightarrow}A$ (or $X_n\rightarrow A$ almost surely) where $A<0$
I want to prove that $Pr[X_n < 0] \rightarrow 1$ (or $X_n < A$ almost ...
1
vote
0answers
20 views
Central Limit Theorem for Dependent Non-Identical Random Variables.
If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$.
How do we apply ...
3
votes
7answers
126 views
How to prove that $1/n!$ is less than $1/n^2$?
I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.
How to prove ...
0
votes
2answers
41 views
An equivalent expression of Cauchy Criterion?
For a sequence $\{a_n\}$, if
$$
\forall \epsilon>0 \ \exists N>0, \forall k \in \mathbf{N}, \ |a_{N+k}-a_N|<\epsilon \
$$
Then $\{a_n\}$ converges and hence is a Cauchy sequence.
Now how ...
0
votes
1answer
30 views
Radius of convergence - ratio test for power series/real numbers
Having trouble with when to apply the ratio test for power series and when to apply the ratio test for real numbers.
For example, find radius of convergence of these....
$\sum_{n=0}^{\infty}(-1)^n ...
1
vote
0answers
23 views
Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?
Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow,
...
2
votes
1answer
18 views
Convergence by using Cauchy Criterion
this is the sequence:
$(a_n)=\frac{1}{n+1}+\frac{1}{n+2}+\cdot\cdot\cdot+\frac{1}{2n}$
And this is what I tried to do so far:
$|a_{n+1} - a_{n} | = \frac{1}{2n+1}+\frac{1}{2n+2}-\frac{1}{n+1} = ...
0
votes
0answers
23 views
Convergence of density function
Let $X_m$ have the density function
$$
f_m(x) = \frac{m}{ \pi(1+m^2x^2)}
$$
where $m \ge 1$. Which modes of convergence have to be respected that $X_m$ converges (if $n \rightarrow \infty$) ?
1
vote
3answers
59 views
Comparison test for series $\sum_{n=1}^{\infty}\frac{n}{n^3 - 2n + 1}$
I am trying to prove the convergence of the series $\sum_{n=1}^{\infty}\frac{n}{n^3 -2n +1}$ with the simple comparison test. I know it can be done with other tests but this question came up in my ...
3
votes
2answers
43 views
Convergence of $\int_0^\infty \sin(t)/t^\gamma \mathrm{d}t$
For what values of $\gamma\geq 0$ does the improper integral $$\int_0^\infty \frac{\sin(t)}{t^\gamma} \mathrm{d}t$$ converge?
In order to avoid two "critical points" $0$ and $+\infty$ I've ...
1
vote
1answer
29 views
Convergence of $\int_0^1 \sqrt[3]{\ln(1/x)} \mathrm{d}x $
Does $$\int_0^1 \sqrt[3]{\ln\left(\frac{1}{x}\right)} \mathrm{d}x$$ converge? WA says it is equal to $\Gamma(4/3)$, however calculating the antiderivative seems approachless to me and can't compare ...
6
votes
1answer
75 views
Existence of a power series converging non-uniformly to a continuous function
I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that:
$f$ converges and is continuous on the closed unit disk $D$ and
the series $\sum_n a_n z^n$ does not converge ...
3
votes
1answer
39 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
3
votes
1answer
31 views
convergence of an integral ( with an inner integral)
I need to figure out for which values of $p \in R $ does the following integral converge?
$$\int_0^{1} \frac{x^p}{\int_0^{x}\ln(1 + \sin(t) + t)dt} {dx} $$
Please note that I don't have to ...
2
votes
1answer
82 views
Borel-Cantelli Lemma
I have some difficulties understanding the following:
Let $(X_n)$ be a sequence of independent random variables s.t.
$P[X_n=1]=1−P[X_n=0]=\frac{1}{n}$
After using the Borell Cantelli lemma, I ...
0
votes
1answer
16 views
Absolute convergence.
Determine if absolutely convergent or not; Justify.
$$\sum_{n=1}^\infty (-1)^n n^2 3^{1-n} x^n \text{ s.t }|x|<3$$
if we take the abs value of $(-1)^n$ we are left with $n^{2} 3^{1-n} x^{n}$ now ...
