Convergence of sequences and different modes of convergence.
0
votes
1answer
24 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
34 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
44 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
26 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
103 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
28 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
67 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
28 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
37 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 $\displaystyle \lim_{b \to \infty} ...
0
votes
2answers
19 views
Develop the next function:$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
59 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:
$$
\begin{align}
x_0 & = a \\
x_1 & = x_0 + \frac{1}{1}(x_0 + x_0(c - 1)) \\
x_2 & = x_1 + \frac{1}{2}(b + x_1(c - 1)) \\
x_3 & = x_2 + \frac{1}{3}(b + x_2(c - 1)) ...
0
votes
1answer
24 views
Convergence of random variable to a negative constant
Let $X_n$ be the sequence of R.Vs and $X_n\overset{P}{\rightarrow}A$ (or $X_n\rightarrow A$ almost surely) where $A<0$
I want to prove that $Pr[X_n < 0] \rightarrow 1$ (or $X_n < A$ almost ...
1
vote
0answers
14 views
Central Limit Theorem for Dependent Non-Identical Random Variables.
If $X_{(1)}, X_{(2)},\ldots$ are mutually dependent as in the case of ordered statistics and we need to find the sum $S_N$ of all $X_{(i)}$ like $\sum_{i=1}^{N\to \infty} X_{(i)}$.
How do we apply ...
3
votes
7answers
124 views
How to prove that $1/n!$ is less than $1/n^2$?
I want to prove
$$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test.
How to prove ...
0
votes
2answers
39 views
An equivalent expression of Cauchy Criterion?
For a sequence $\{a_n\}$, if
$$
\forall \epsilon>0 \ \exists N>0, \forall k \in \mathbf{N}, \ |a_{N+k}-a_N|<\epsilon \
$$
Then $\{a_n\}$ converges and hence is a Cauchy sequence.
Now how ...
0
votes
1answer
29 views
Radius of convergence - ratio test for power series/real numbers
Having trouble with when to apply the ratio test for power series and when to apply the ratio test for real numbers.
For example, find radius of convergence of these....
$\sum_{n=0}^{\infty}(-1)^n ...
1
vote
0answers
23 views
Is $\phi^T_tP_t^{-1}\phi_t\to 0$ when $P_{t+1}=\sum_{k=0}^t\phi_k\phi_k^T+P_0$?
Let $\phi_t\in\mathbb{R}^n$, $\forall t\geq0$, and $\sup_t\|\phi_t\|_2^2\leq M<\infty$(euclidean norm). Define $n\times n$ positive definitive matrices as follow,
...
2
votes
1answer
18 views
Convergence by using Cauchy Criterion
this is the sequence:
$(a_n)=\frac{1}{n+1}+\frac{1}{n+2}+\cdot\cdot\cdot+\frac{1}{2n}$
And this is what I tried to do so far:
$|a_{n+1} - a_{n} | = \frac{1}{2n+1}+\frac{1}{2n+2}-\frac{1}{n+1} = ...
0
votes
0answers
18 views
Convergence of density function
Let $X_m$ have the density function
$$
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$$ converge?
In order to avoid two "critical points" $0$ and $+\infty$ I've ...
1
vote
1answer
28 views
Convergence of $\int_0^1 \sqrt[3]{\ln(1/x)} \mathrm{d}x $
Does $$\int_0^1 \sqrt[3]{\ln\left(\frac{1}{x}\right)} \mathrm{d}x$$ converge? WA says it is equal to $\Gamma(4/3)$, however calculating the antiderivative seems approachless to me and can't compare ...
6
votes
1answer
72 views
Existence of a power series converging non-uniformly to a continuous function
I am wondering whether there exist a function $f(z) = \sum_{n\geq0} a_n z^n$ such that:
$f$ converges and is continuous on the closed unit disk $D$ and
the series $\sum_n a_n z^n$ does not converge ...
3
votes
1answer
39 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
3
votes
1answer
30 views
convergence of an integral ( with an inner integral)
I need to figure out for which values of $p \in R $ does the following integral converge?
$$\int_0^{1} \frac{x^p}{\int_0^{x}\ln(1 + \sin(t) + t)dt} {dx} $$
Please note that I don't have to ...
2
votes
1answer
79 views
Borel-Cantelli Lemma
I have some difficulties understanding the following:
Let $(X_n)$ be a sequence of independent random variables s.t.
$P[X_n=1]=1−P[X_n=0]=\frac{1}{n}$
After using the Borell Cantelli lemma, I ...
0
votes
1answer
16 views
Absolute convergence.
Determine if absolutely convergent or not; Justify.
$$\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 ...
1
vote
1answer
19 views
Equicontinuity and uniform convergence 2
Let $\{f_n\}_n$ be a sequence of real valued functions on a compact metric space $K$. Suppose that for all $x$ we have $f_n(x) \to f(x)$ as $n \to \infty$ and that the family $\{f_n\}_n$ is ...
1
vote
1answer
45 views
Suppose E is an infinite subset of a metric space X.
Prove that x is a limit point of $E$ if and only if there is a sequence $\left \{ x_n \right \}^\infty_{n=1} \subset E $ that converges to x.
This was part of our practice final and I have no idea ...
1
vote
0answers
37 views
stable, consistent and convergence of one step method
the one step method defined by
$x_{k+1} = x_k + hγ(t_k,x_k)$
for the ODE
$dx(t)/dt = f(t,x(t))$
with
$γ(t_k,x_k) = f(t_k+h/2,x_k+(h/2)*f(t_k,x_k))$.
what is the conditions for the ...
2
votes
1answer
29 views
pointwise convergence implies $L^p$ convergence in this case?
If $f_n(t) \to f(t)$ pointwise and $\int_0^T f(t)$ is finite, does $f_n$ converge to $f$ in $L^p$ for any $p$? I think so, because $f_n$ converges so it's bounded, so one can use DCT. Am I right?
0
votes
0answers
31 views
Does $f(X_n)\to f(X)$ in probability imply $X_n\to X$ in probability?
Does $f(X_n)$ converge in probability to $f(X)$ imply $X_n$ converge in probability to $X$?
2
votes
3answers
28 views
behavior of a sequence
Imaging a sequence $ a_{k} \in \Omega $ with $ \Omega \subset \Bbb{R} $ closed, $ \lim\limits_{k \to \infty} \| a_{k+1} - a_{k} \| = 0 $.
My Professor said that because of this the sequence would ...
0
votes
1answer
25 views
Brownian motion and convergence in probability of step functions
For positive $a$ and Brownian motion $B$, I want to compute $\int_0^a g(s)dB_s$ where $g \in L^2$ and $g$ is a step function if there exists partition $0=t_0 < ... < t_n = a$ such that $g = ...
3
votes
3answers
51 views
Determine if a sequence converges using the number e
Knowing that the number $e =\lim_{n\to\infty}\left(1+{1\over{n}}\right)^n$ solve $a_n=\left({n+1}\over{n+3}\right)^n$
So (...)
$$\lim_{n\to\infty}\left({n+1}\over{n+3}\right)^n = ...
0
votes
1answer
17 views
Dose X converges in probability to Y converges in probability to a constant z implies X converges in probability to z
Suppose we have $\frac{1}{n}\sum_j^n X_{ij}$ converges in probability to $Y_i$ and $\frac{1}{n}\sum_y^n Y_{j}$ converges in probability to a constant $z$, where $Y_i$ is not the expectation value of ...
1
vote
1answer
45 views
Cauchy multiplaction
Would like to know if exists an example for $$\sum_0^\infty a_n x^n,\sum_0^\infty b_n x^n$$ $$\sum_0^\infty c_n x^n, c_n:=\sum_{k=0}^n a_k b_{n-k} $$
such that $\max\{R_a,R_b\} < R_c < \infty$ ...
3
votes
1answer
33 views
Almost sure convergence problem
I'm working on a problem in which I should prove "almost sure" convergence for a sequence of random variables. I'm using Borel-Cantelli lemma to prove it. Here is the question and my solution - I ...
1
vote
2answers
39 views
Testing Convergence of Integral
How do I prove that the integral: $\int_0^1 x^p{} (1-x)^q dx$ converges or diverges?
2
votes
1answer
35 views
Question on Convergence in Probability
I appreciate if you could guide me on this question:
Assumptions:
$X_n \rightarrow^p c$: $X_n$ convrges in probability to a constant c.
g(.) is any function that satisfies:
$$\text{if } a_n - c = ...
2
votes
2answers
32 views
What's the radius of convergence of the next sum: $\sum_{n=0}^\infty (\int_o^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt)x^n$
What's the radius of convergence of the next sum: $$\sum_{n=0}^\infty \left(\int_0^n\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt\right)x^n$$
I know that $$\int_0^\infty\frac{\sin^2t}{\sqrt[3]{t^7+1}}dt$$ does ...
0
votes
0answers
99 views
Qualities of Projected Gradient Methods
Consider the following constrained minimization problem:
$ min_{x \in X} \ f(x) $
where $ X \subset \Bbb{R}^{n} $ is a nonempty closed convex set and f is continuously diferentiable.
I'm ...
2
votes
1answer
66 views
convergence and divergence of a series
Let {$a_n$} be a sequence of non-negative real numbers such that the series $\sum^\infty_{n=1} {a_n}$ is convergent. If $p$ is a real number such that the series ...
1
vote
1answer
49 views
Conditions for taking a limit into an infinite sum
Suppose $f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}g_{n}\left(x\right)}$ under what conditions is it true that: $$\lim_{x\to c}f\left(x\right)={\displaystyle \sum_{n=0}^{\infty}\lim_{x\to ...
0
votes
2answers
34 views
Series convergence with exponentials
I would like to understand if the following series converge (any closed form for that?!):
$$\sum_{n=0}^{\infty}\quad \frac{\exp(-n\cdot a)+n\cdot b}{(n+1)^2}$$
$$\sum_{n=0}^{\infty}\quad ...
2
votes
1answer
37 views
Radius of convergence of a power series with Bernoulli numbers
Say, we use the definition:
Bernoulli numbers arise in Taylor series in the expansion $$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k \frac{x^k}{k!}$$
and then derive power series representations of the ...
0
votes
1answer
31 views
Convergence of sequence uneven/even
Given is:
$
(a_n )_{n=1}^x $ with (x = infinite)
and with
$ a_n = \frac{1*3*...*(2n-1)}{2*4*6...*(2n)} $
I have to show if the sequence is convergent or not:
I thought about showing that the ...
