Convergence of sequences and different modes of convergence.

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3
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2answers
37 views

Hints on showing Cauchy sequence converges

Let $T>0$ and $L\geq0$. Let $C[0,T]$ be the space of all continuous real valued functions on $[0,T]$ with the metric $\rho$ defined by $$\rho(x,y)=\sup_{0\leq t\leq T}e^{-Lt}\left|x(t)-y(t)\right|$...
4
votes
1answer
34 views

Find the radius of convergence of this power series: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$

Given: $\sum_{k=0}^{\infty } \binom{2k}{k}x^{k}$ I started by forming it: $\binom{2k}{k} = \frac{(2k)!}{k!*(2k-k)!} = \frac{(2k)!}{k!*k!}$ Now the problem is, I cannot write $2! * k!$ instead of $(...
3
votes
2answers
51 views

Improper integral - checking convergence of $\int_{1}^{\infty} x^2 \sin(x^4) dx$

Does the following improper integral converges ? $$\int_{1}^{\infty} x^2 \sin(x^4) dx$$ Tried to find some known improper integral to compare this one to, but didn't find one. Thanks for helping!
2
votes
3answers
27 views

Exponential limit convergence for each $x$

I have $f_n(x)=\left( 1+\frac{-e^{-x}}{n} \right)^n$, what about the convergence to $f(x)=e^{-e^{-x}}$? is it true $\forall x?$ I say yes, but how can I show this? Is continuity of $f_n$ enough?
0
votes
1answer
39 views

Does this converge?

If I have $$X_i=\begin{cases}2\quad p=\frac{1}{3}\\ \frac{1}{2}\quad p=\frac{2}{3} \end{cases}$$ random variables with the same distribution. How can I compute the limit almost sure as $n\to\infty$ ...
-1
votes
1answer
28 views

Problem involving sequence of random variables on probability space [closed]

How do I construct (and prove that) an example of a sequence of random variables $\{X_n\}_{n\, \ge\, 1}$, on an appropriate probability space, for which $X_n$ converges to $0$ in $L^r$ for all $r > ...
1
vote
1answer
42 views

Convergence of $\sum_{n=0}^\infty n^{1/n}-1$ and $\sum_{n=0}^\infty (1/n!)^{1/n}$

$$\sum_{n=0}^\infty n^{1/n}-1$$ $$\sum_{n=0}^\infty (1/n!)^{1/n}$$ Hi. I am working on calculus now. While studying convergence test part, I ran into those problems... Wolfram alpha says they both ...
0
votes
0answers
30 views

The connection between the length of Fibonacci $p$-step numbers and it's limit values

One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows \begin{equation}\label{cp26} F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...
0
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0answers
20 views

problem in convergence and uniform convergence of sequence of functions?

Let $f: \mathbb{R} \to [0, \infty)$ be a non- negative real-values continuous function. Let $$ \phi_n(x) = \begin{cases} n, \ \quad if \ f(x)\geq n\\ 0, \ \quad if \ f(x) < n\end{...
1
vote
1answer
43 views

Interval of convergence for series with complex numbers

I'm trying to find interval of convergence of this series: $$\sum_{n=1}^{\infty} \frac{7^n(z+2i)^n}{4^n+3^ni}$$ and I should draw a plot which represents the answer, this is what I've got so far: ...
0
votes
3answers
81 views

Convergence problem $\sum \left(1-n\sin\left(\frac{1}{n}\right)\right)$ [closed]

I have to check convergence of: $$\sum_{n=1}^\infty\left(1-n\sin\left(\frac{1}{n}\right)\right).$$ I have no idea but I only check that $\lim \ n\left(1-n\sin\left(\frac{1}{n}\right)\right)=0$.
2
votes
0answers
35 views

Convergence of sequence with $\zeta$ function

Last time I heard interesting question. Unfortunately I do not have idea how to solve it, so I decided to give it here. Let us define sequence $a_n=(\underbrace{\zeta\circ...\circ \zeta}_{n})(\pi)$ ...
2
votes
3answers
80 views

Convergence of $\sum a^{1/x_n}$ for $a$ in $(0,1)$ and $\sum x_n$ a positive convergent series

Let $\sum x_n$ be a convergent series of positive real numbers and $0<a<1 $, then is the series $\sum a^{1/{x_n}}$ convergent ? I have only figured out that $\lim a^{1/{x_n}}=0$.
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0answers
20 views

Showing $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ is a Hilbert space

Let $I$ be an open interval in $\mathbb{R}$. We define $H= \{v \in H^1(I) \ | \ v(0)=0 \} \subset H^1(I)$ with the scalar product of the Sobolev space $H^1(I)$, i.e. $(u,v)=(u,v)_{L^2(I)}+(u',v')_{L^...
0
votes
3answers
59 views

convergence of $\int _0^1\:\frac{\ln\left(1-x\right)}{\left(1+x\right)^2}dx$

How do I prove convergence of $$\int _0^1\:\frac{\ln\left(1-x\right)}{\left(1+x\right)^2}dx$$ and if it's convergent, calculate the value of the integral? I noticed that the values that the function ...
0
votes
1answer
24 views

Radius of convergence of power series of $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ at $0.$

The function $f(z) = \frac{z^{3}-1}{z^{2}+3z-4}$ has a power series expansion in a neighborhood of the origin. What is its radius of convergence. I believe I have to use the ratio test and show that ...
2
votes
1answer
40 views

Where is the mistake of a possible application of Frullani's theorem in this case?

My question is about what is the problem, if there is one, to get an identity using Frullani's integral. I've in a hand the statement from MathWorld, and other statement from this site, with a nice ...
1
vote
1answer
37 views

Convergence of the integral: $I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx$

Study the convergence of the integral: $$I_{\alpha }=\int _0^{\infty }\left(\frac{e^{-\alpha x}}{\sqrt{x}}\right)\:dx$$ and calculate $I_2$. Ok so to study the convergence I'm using convergence ...
1
vote
2answers
39 views

An example of a power series that has a radius of convergence of 3

The problem states "Give an example of a power series $\sum^{\infty}_{n=0}$a$_{n}$z$^{n}$ that has a radius of convergence of 3 and that represents an analytic function having no zeroes. I'm sorry if ...
0
votes
2answers
44 views

How to show the sequence is monotone

"$u_n = \frac{2}{1+e^{-n}}$. Show that $u_n$ is monotone." My approach would be to consider |$u_{n+1} - u_n$| = |$\frac{2}{1+e^{-n-1}} - \frac{2}{1+e^{-n}}$|. However I'm not sure the best way to ...
2
votes
1answer
43 views

Is this proof of convergence in probability correct?

${X_i}, i = 1,2,\dots$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $\frac{S_n}{n} \to 0 \quad $ in probability show that $$\lim_{n\to \infty} \min_{...
0
votes
1answer
23 views

Convergence/Divergence of Integral, can P-test be used here?

I have an integral like this: How do I check its convergence? As far as I know, P-test can be used for integrals from 0 to 1, or A to infinity, what would I do in this case?
1
vote
1answer
23 views

Show that MLE estimator convergences in probability to actual parameter

For iid stochastic variables $X_1, ..., X_n$ whose distribution is defined by 2 parameters, I have found the MLE estimators. They are $\hat{\mu} = \sum x_i/n$, and $\hat{\lambda}$ given by $$ \frac{...
0
votes
0answers
19 views

Different radius of convergence for ratio test and Hadamard's formula

I'm pretty sure I'm missing something very basic... But I have the following question: Determine the radius of convergence of $\sum \alpha_n z^n$ with $\alpha_n=\frac{1}{n+1}$. Now, with the ratio ...
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0answers
14 views

A false identity involving $2^{\frac{1}{\zeta(s)}}$ for $\Re s>1$, from these particular values of the Riemann Zeta function and its alternating

Yesterday when I was exploring symbolic calculations $\dagger$ about specializations in $z=\frac{1}{n}$ with $n>1$ an integer, of $$\zeta(z)=(1-2^{1-z})^{-1}\sum_{n=1}^\infty\frac{(-1)^{n-1}}{n^z}:=...
1
vote
1answer
49 views

A convergent series implies two split convergent series

Consider the series $$ \sum_{n=0}^{\infty}\left ( \frac{1}{n+1}-\frac{1}{z+n} \right )\tag{1} $$ It converges for all $z\notin \{0\}\cup \mathbb{Z}^-$. Does it imply, that $$ \sum_{n=0}^{\infty}\frac{...
2
votes
1answer
37 views

An ancillary result from convergence in probability

I was reading a paper concerning probability theory. We have that $X_i$, $i = 1,2,...$ i.i.d random variables, and $S_n = \sum_{i=1}^nX_i$ is defined as partial sum as usual. If $$\frac{S_n}{n} \...
2
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0answers
73 views

Question about an infinite product

The following infinite product is well known: $ (1) \frac{\sqrt{1-\alpha^2}}{\arccos \alpha} = \frac{\sqrt{2+2\alpha}}{2}\frac{\sqrt{2+\sqrt{2+2\alpha}}}{2}\frac{\sqrt{2+\sqrt{2+\sqrt{2+2\alpha}}}}{2}...
4
votes
3answers
123 views

Does $\int_0^2\frac{1}{\ln(x)}dx$ converge?

Given the following integral: $$ \int_0^2 \frac{1}{\ln(x)} dx $$ Does it converge? Iv'e gone this far: $$ \int_0^2 \frac{1}{\ln(x)} dx = \int_0^1 \frac{1}{\ln(x)} dx + \int_1^2 \frac{1}{\ln(x)} dx $...
1
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0answers
23 views

Convergence of $\sum_{n=1}^\infty\frac{\psi(n)}{e^n}\sin ns$ on an horizontal closed strip

Let $\psi(x)=\sum_{k\leq x}\Lambda(k)$ the Second Chebyshev function, and $\epsilon>0$. I would like to ask Question. Can you prove or disprove that the series $$\sum_{n=1}^\infty\frac{\psi(n)}{...
0
votes
2answers
29 views

Convergence of integral with absolute function

Given that the following integral converges: $$ \int_{0}^\infty |f(x)| dx$$ Prove or disprove that the following integral also converges: $$ \int_{0}^\infty f(x) dx$$ I thought to use the squeeze ...
1
vote
1answer
34 views

Fourier series for $\min \{0, \cos x\}$

How can I find Fourier series and convergence of function $f(x)= \min \{0, \cos x \}$ ? Because it is an even function I am expanding it only for cosine. Doing that I get $a_{0} = 0$, after that I ...
1
vote
1answer
38 views

Convergence of real numbers along an ultrafilter

Suppose we have an uncountable set $I$ and a non-principal ultrafilter $U$ on it. I am interested whether it is possible to conclude that if cardinality of $I$ is big enough, then every tuple $(x_i)_{...
4
votes
2answers
67 views

$\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2+n}$

Does the following series converge or diverge $$\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2+n}$$ I know that $\sum_{n=1}^{\infty}\frac{1}{(\ln{n})^2}$ diverges. $\sum_{n=1}^{\infty}\frac{1}{n(\ln{n})^2}$ ...
0
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0answers
63 views

How much can the integrability at zero tell about the decay rate around zero?

Suppose that $g$ is a continuous, nonincreasing and nonnegative function on $(0,1)$. The question is whether one can characterize the integrability of such functions at zero by their decay rates at ...
0
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0answers
19 views

How to find similar convergence rates?

Consider the Taylor's series infinite summation of $\sin(x)$. Let $A_k=\sum\limits_{i=0}^k(-1)^i{x^{2i+1}\over (2i+1)!}$ (Series expansion of $\sin(x)$) I need a series $\{C\}_n$such that its ...
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vote
0answers
22 views

Newton method and convergence

$f(x,y)=\begin{pmatrix}x^2-y^2+1\\ 2xy\end{pmatrix}=\begin{pmatrix}0\\0\end{pmatrix}$ and $(x_0,y_0)=(1,1)$ Do the first 4 steps of Newton method. I did this. Show that are many reasonably starting ...
2
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1answer
43 views

Proving monotonic decreasing of general term: $ \sum_{n=2}^\infty (-1)^{n+1}\dfrac{n^2-1}{n^3-1} $

I have got the following series - $$ \sum_{n=2}^\infty (-1)^{n+1}\dfrac{n^2-1}{n^3-1} $$ I know that this an alternating series which converge - but got confused on how to prove this the ...
1
vote
1answer
32 views

Does $X_n \xrightarrow{L_1} X \implies X_n \xrightarrow{\text{qm}} X$?

Let $X_n$ and $X$ be a sequence of random variables. According to All of Statistics (pg. 81), we have that: $$ X_n \xrightarrow{\text{qm}} X \implies X_n \xrightarrow{L_1} X $$ But the book doesn't ...
0
votes
1answer
20 views

Right-const function and pointwise/uniform convergence

Let function $f:\mathbb{R} \rightarrow \mathbb{R}$ be right-const iff $\exists_{M \in \mathbb{R}}\forall_{x,y \ge M}f(x)=f(y)$. Consider function sequence $\{f_n\}$ which every term is right-const. ...
0
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0answers
24 views

$L^p$ convergence of smooth compactly supported functions

I already checked the similar question here, but want to check slightly different argument. Given $f(x)\in L^p(\mathbb{R}^n)$, can I find $f_n(x) \in C^\infty_0$ s.t. $f_n \rightarrow f$ almost ...
1
vote
1answer
31 views

What should be the value of $\alpha$ for which the series is convergent?

The series $$\sum \frac{\log(1+\frac{1}{n})}{n^\alpha}$$ a. Converges if $\alpha>0$ b. Diverges for all $\alpha\in \mathbb{R}$ c. Converges if $\alpha=0$ d. Converges if $\alpha<0$ ...
3
votes
2answers
60 views

Is $t_n=\frac{1}{n} \left(1+\frac{1}{\sqrt2}+\cdots+\frac{1}{\sqrt n}\right)$ convergent?

Let $$t_n=\frac{1}{n} \left( 1+\frac{1}{\sqrt2}+\cdots+\frac{1}{\sqrt n} \right)$$ Is the following series convergent? If we let $a_n=\frac{1}{\sqrt n}$, then $$\lim_{n\to\infty}a_n=0\rightarrow \...
1
vote
2answers
64 views

Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem

Show that the Maclaurin series of $f(x)=1/\sqrt{1-x}$ holds for $x\in[0,1/2]$ using the Lagrange remainder theorem I can see that $$f'(x)=\frac12 (1-x)^{-\frac32}\text{ and }f''(x)=\frac12\frac32(1-...
0
votes
1answer
45 views

(Conceptual) Continuity of binary relation $\succsim$ and definition using contour sets

Some background information: $\succsim$ is a binary relation that represents preference between two goods. $\succsim$ means "x is at least as good as y." Continuity of this relation is defined to be ...
0
votes
1answer
80 views

Examining a solution of a differential equation without knowing the solution

The differential equation is given by $$\dot x=-x \cos x$$ with $x(0)=x_0\in(0,\frac{\pi}{2})$. Now I need to show that for each choice of $x_0$ the domain of the solution $x: I\rightarrow \mathbb{...
0
votes
1answer
27 views

weakly convergence the sequence $f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]}$

I need to research on the uniform, weak and strong convergence the sequence $$f_{n}= n. \chi_{[-\frac{1}{n},\frac{1}{n}]}$$ for $n\in \mathbb{N},$ in $L^{2}(\mathbb{R})$ equipped with norm $\...
2
votes
1answer
101 views

For what values of $a$ does $\int_0^\infty\left(\frac{x^a}{1 + x^2}\right)^4 \, dx$ converge?

I'm learning about convergence/divergence of improper integrals and need help with the following problem: Find for what values of $a$ does the following integrals exists $$(1) \int_0^\infty\...
2
votes
2answers
36 views

Comparision test for this series?

How do I check divergence of this series? $$\sum_{n=0}^{\infty} \frac{6}{4n-1} - \frac{6}{4n+3}$$ Wolframalpha said it used the comparision test but I don't see what possible smaller sum to use? ...
2
votes
1answer
54 views

conditional convergence of $\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$

prove that the series $$\sum_{n=2}^{\infty} \frac{\cos(n)}{n}$$ is conditionally convergent? I tried to prove that it is not absolutely convergent series by trying to prove that $\sum_{n=2}^{\infty} \...