Convergence of sequences and different modes of convergence.

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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 ...
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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 ...
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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
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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
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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} ...
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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 ...
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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 ...
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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 ...
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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}}$ ...
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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} ...
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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 ...
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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 ...
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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
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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
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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 ...
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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
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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 ...
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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 ...
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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 ...
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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
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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} = ...
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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 $n \rightarrow \infty$) ?
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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 ...
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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 ...
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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?
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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
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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 ...
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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 = ...
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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 = ...
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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 ...
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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
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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 ...
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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?
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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 = ...
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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 ...
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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
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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 ...
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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
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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
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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
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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 ...

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