Convergence of sequences and different modes of convergence.

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4
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3answers
96 views

Prove $(x_n)$ defined by $x_n= \frac{x_{n-1}}{2} + \frac{1}{x_{n-1}}$ converges when $x_0>1$

$x_n= \dfrac{x_{n-1}}{2} + \dfrac{1}{x_{n-1}}$ I know it converges to $\sqrt2$ and I do not want the answer. I just want a prod in the right direction. I have tried the following and none have ...
0
votes
0answers
13 views

Describe the characteristics of this function (where it is defined, continuous, diff, twice diff).

Consider the function $f(x)=\sum\limits_{k=1}^{\infty}\frac{\sin(x/k)}{k}$ Where is $f$ defined? Continuous? Differentiable? Twice-differentiable? My thoughts so far: Initially I thought that due ...
-1
votes
0answers
33 views

How to show norm convergence of function series ?

Let $\mathcal{C^1([0,1])}$ be the vector space of continuously differentiable functions on $[0,1]$ and let $f_n$ be a function series. Let $||f||_1=\int_{0}^1 f(x) dx$ and ...
3
votes
2answers
48 views

Convergence of a series involving arccot?

I am supposed to determine the convergence (conditional or absolute) or the divergence of the series $$\sum_{n=1}^\infty \frac{\cot^{-1} ({\pi/4 + n\pi})}{\sqrt{3n^3+1}}$$ I previously solved a ...
4
votes
2answers
116 views

Convergence of the series $\sum\limits_n\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}} $

Does the following series converge? $$\sum_{n=1}^{\infty} \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n \text{ th}}}\ $$
2
votes
1answer
16 views

Necessary and sufficient conditions for equivalence of different types of convergences for Bernouli RVs

Let ${X_n}_{n=1}^{\infty}$ be a sequence of independent random variables such that $Pr(X_n = 1) = p_n$, $Pr(X_n = 0) = 1 - p_n$, where ${p_n}_{n=1}^{\infty}$ is a sequence of numbers in $[0, 1]$. (a) ...
1
vote
1answer
29 views

A sufficient condition for convergence in probability

Let $X_n$ $(n \geq 1)$ and $X$ be real-valued random variables defined on the same probability space. If every subsequence of $X_n$ contains a further subsequence that converges to $X$ almost surely, ...
0
votes
1answer
49 views

Determining the order of convergence of $ X_{n+1} = \frac{(X^3_n + 3aX_n)}{(3X^2_n + \alpha)} $

I need to find the order of convergence for: $$ X_{n+1} = \frac{(X^3_n + 3aX_n)}{(3X^2_n + \alpha)} $$ In a previous part we are told $\alpha$ = 2 and $x_0$=1. I know the first step is to take the ...
1
vote
1answer
19 views

Show Newton's iteration to compute the root $x^*$ of $|x|^{\frac{3}{7}}$ does not converge

I need to show that Newton's iteration to compute the root $x^*$ of $$f(x) = |x|^{\frac{3}{7}}$$ does not converge for any starting guess $x_0 \neq 0$. The first thing I did was to create the ...
0
votes
2answers
42 views

Study the Convergence of a Series

For a sequence ${a_n}$ that is monotonically decreasing towards $0$, study the convergence of the series: $$\sum_{n=1}^\infty (-1)^{n+1}\frac{a_1 + a_2 + a_3 + \cdots + a_n}{n}$$ In summary, does ...
1
vote
2answers
66 views

Finding the limit of lim${_{n \rightarrow \infty}}\left( \dfrac{n^3}{2^n} \right)$

For a class of mine we were given a midterm review; however, I just cannot figure out how to finish this one: Finding the limit of lim${_{n \rightarrow \infty}}\left( \dfrac{n^3}{2^n} \right)$ My ...
0
votes
1answer
21 views

Multiple sequences of random variables that converge in probabilty

I'm struggling with this exercise: For each $k\in \mathbb{N}$, let $(X^{(k)}_n)_{n\in\mathbb{N}}$ be a sequence of real random variables converging to $0$ in probabilty as $n\to\infty$. Define for ...
3
votes
0answers
84 views

On the Laplace transform $\int_0^\infty e^{-sx}d \left( \ \int_2^{e^{1+x}}\frac{dt}{\log t}\right) $

I've read the basics about Laplace transform, and I know that since for $\Re s>1$, $\frac{e^x}{1+x}$ has exponential order, then $$F(s)=\int_0^\infty e^{-sx}\frac{e^{1+x}}{1+x}dx$$ is well defined, ...
0
votes
0answers
14 views

Marginal convergence in distribution plus independence imply joint convergence?

Suppose $X_n \stackrel{d}{\to} X$, $Y_n \stackrel{d}{\to} Y$, and $X$ and $Y$ are independent. Does it follow that $(X_n, Y_n) \stackrel{d}{\to} (X,Y)$? I don't think this is true, but am having ...
0
votes
1answer
38 views

The Cauchy product $\sum_{n=1}^\infty \frac{\log n}{e^n}= \left( 1-\frac{1}{e} \right)\sum_{n=1}^\infty\frac{\log n!}{e^n} $

I know that the Cauchy product is defined $$\left(\sum_{n=1}^\infty\frac{\log n}{e^n}\right)\left( \sum_{n=1}^\infty\frac{1}{e^n} \right)= \sum_{n=1}^\infty\sum_{k=1}^n\frac{\log k}{e^{k+n-k+1}},$$ ...
0
votes
3answers
86 views

Prove or Disprove the convergence of a series [closed]

Let the following series - \begin{equation} \sum_{n=1}^{\infty}\frac{(2n)!}{(4^n)(n!)^2(2n+1)}x^{2n+1} \end{equation} $$x>0 , x\ne1 $$ Well I tried to prove that for x < 1 this series ...
1
vote
1answer
34 views

On $-\frac{\zeta'(x)}{x\zeta(x)}$ and von Mangoldt function

I believe that it is possible show the following Fact. For real $x>e$ then $$-\frac{\zeta'(\log x)}{x\zeta(\log x)}=\sum_{n=1}^\infty\frac{\Lambda(n)}{n^{\log x}},$$ where $\zeta(x)$ is the ...
3
votes
3answers
34 views

Show that $A$ and $A^C$ are both dense in $(\ell^2,\lVert \cdot \rVert_2)$, where $A=\{x\in\ell_2:\sum_{k=1}^\infty x_k\neq0\}$.

The title says it all. Showing $A$ is dense in $\ell_2$ seems easy; for any $x\notin A$, for each $n\in\mathbb N$ let $x^n$ in $\ell_2$ where $x^n$ is identical to $x$ except that $x^n_1=x_1 + ...
2
votes
1answer
69 views

The doubly infinite series $\sum_{-\infty}^{+\infty} n$

I have the following question from Function Theory of One Complex Variable - Greene/Krantz: Give an example of a series of complex coefficients $ a_n$ such that $\lim_{N \to + \infty} \sum_{n= ...
1
vote
2answers
75 views

Prove/Disprove the convergence of series [closed]

Hey guys I would like to get some hint for this series - I tried some test but couldn't decide if the series converge or diverge - $$\sum_{n=2}^\infty ...
10
votes
0answers
84 views

Are the fractional parts of powers of $\pi$ divergent?

Let us define $a_n$ as the fractional part of $\pi^n$. In other words, define $a_n=\pi^n-\lfloor \pi^n \rfloor$. Then, does the following limit exist? $$\lim_{n \to \infty}a_n$$Intuitively, it ...
2
votes
2answers
51 views

Convergence of $\sum_{i \leq n} X_i/n$

I have a question like this: Let $(X_n)$ be an i.i.d sequence of random variables with values in $\{-1,1\}$, and define $Y_n:= \sum_{i \leq n} X_i/n$. Show that $(Y_n)$ converges almost surely and in ...
0
votes
0answers
31 views

Unbounded, convergent sequence

According the the Monotone Convergence Theorem, a monotone sequence is convergent if and only if it is bounded. I started thinking about a sequence that at first sight, appears to break that rule. ...
1
vote
1answer
18 views

$x \in \bar A$ iff $\exists (x_\lambda) \subset A$ net with $(x_\lambda) \to x$

$x \in \bar A$ $\iff$ $\exists (x_\lambda)_{\lambda \in \Lambda} \subset A$ net with $(x_\lambda) \to x$ What I did: $\Leftarrow$: Let $(x_\lambda) \subset A$ a net with $(x_\lambda) \to x$. Let $V$ ...
1
vote
1answer
35 views

Weierstrass uniform convergence - Stuck to the point.

$$\sum_{n=1}^\infty \frac{\sin (nx)}{n!}$$ Interval: $x \in(- \infty, + \infty)$ I've been trying to do this all day, but I just cant get to the end of it. It's not that I do not understand the ...
2
votes
1answer
45 views

Prove/Disprove that the series is converge

I have the following series - $$\sum_{n=2}^\infty {\frac{\sin(1/n)}{\ln(n)}}$$ Well , I tried some of the tests for series but didn't succeed to get to the answer. Thanks!
2
votes
1answer
32 views

$T$ can be $\infty$ with positive probability

From Williams' Probability with Martingales How exactly do we know $T$ can be $\infty$ with positive probability or $$P(T = \infty) > 0 \text{ ?}$$ I'm guessing that that means there ...
1
vote
0answers
32 views

When convergence a.s. implies convergence in mean?

Can someone help me with proving the following: Assume that $X_n$ converges almost surely to $X$, where $X_n$ is a sequence of non-negative random variables. Furthermore, assume that the sequence ...
0
votes
1answer
31 views

Prove that if the series converges then the inside sequence is bounded by this specific number

Let $$a_n = \sum_{k=0}^n b_k$$ converge, then for some $N$ sufficiently large: $$|b_n| \le \frac{1+\max\{|b_1|, |b_2|,...|b_N|\}}{2}$$ If the series converges that means $b_n$ converges to $0$. ...
2
votes
2answers
64 views

Find the interval of convergence $\displaystyle \sum_{n=1}^\infty \dfrac{(3n)!(x^n)}{(n!)^3}$

$\displaystyle \sum_{n=1}^\infty \dfrac{(3n)!(x^n)}{(n!)^3}$ I already found the interval of convergence to be $\displaystyle -\frac{1}{27} < x < \frac{1}{27}$. I am having trouble checking the ...
2
votes
1answer
38 views

Prove that : $X_n \xrightarrow{\mathrm{a.s.}}0\iff \sum_n P(X_n>0) <\infty$

Let $(X_n)$ be a sequence of independent integer-valued (nonnegative integers) random variables Prove that $X_n\xrightarrow{\mathrm{a.s.}} 0\iff \sum_n P(X_n>0) <\infty$ For the ...
6
votes
1answer
73 views

Prove series converge for almost every $x$

Let $f\in L^p(\mathbb{R})$, $1<p<\infty$, and let $\alpha>1-\frac{1}{p}$. Show that the series $$\sum_{n=1}^{\infty}\int_n^{n+n^{-\alpha}} |f(x+y)|dy$$ converges for a.e. $x\in ...
0
votes
3answers
44 views

Show pointwise convergence and (potentially) uniform convergence $\sum_{k=1}^\infty\frac{x^k}{k}$

I am looking to show pointwise convergence and (potentially) uniform convergence of the following: $$\sum_{k=1}^\infty\frac{x^k}{k}$$ I know (from my book) this converges for my given values of $x ...
0
votes
0answers
26 views

Find where {fn(x)} n=1 to infinity converges pointwise

fn(x) = $ \frac{x}{(1+x)}$ and x ∈ R Find all real numbers x where fn(x) converges pointwise and describe the limit function. Any hints or suggestions?
2
votes
2answers
17 views

$X_n\to 0$ in probability implies $E[f(X_n)]\to f(0)$ for $f$ uniformly cts and bounded

Let $f$ be a uniformly continuous and bounded function. I've shown that if $X_n\to 0$ in probability, then $f(X_n)\to f(0)$ in probability as well. Now I want to say that $$\lim_{n\to\infty} ...
0
votes
1answer
61 views

Does a convergent sequence in theory ever reach its limit?

Completing a question on the sequence $\{a_n\} = \frac n{2n+1}$. Does $a_n$ in this sequence ever actually get to $\frac12$ officially?
1
vote
1answer
19 views

Sequences of random variables converging in probability to the same limit a.s.

Let $(X_n)_{n \geq 1}$ and $(Y_n)_{n \geq 1}$ be two sequences of random variables s.t. $X_n$ converges to X and $Y_n$ to $Y$ both in probability. Furthemore, $X$ = $Y$ a.s. How can I prove that, ...
2
votes
1answer
30 views

Uniform convergence on compact subset

Let there be two functional squences $$a_n(x)=\sqrt[n]{x} \quad \textrm{ for $x\in(0,\infty)$}$$ $$b_n(x)=\sum_{k=0}^{n}x^k(1-x)^k=\frac{1-x^{n+1}(1-x)^{n+1}}{x^2-x+1} \quad \textrm{ for $x\in ...
2
votes
1answer
33 views

$P($partial sums of $\sum X_k$ are bounded$)>0 \to \sum_{k=1}^{\infty} X_k < \infty\ \text{a.s.}$

From Williams' Probability with Martingales How is the remark deduced from the proof of $b$? I really don't see it.
0
votes
1answer
29 views

Uniform and pointwise convergence

So we had the pointwise and uniform convergence, and I do get that a sequence of function can converge to a function, just like ordinary sequences do. But what I don't quite get is this pointwise and ...
0
votes
1answer
49 views

If $M_n \to M_{\infty}$ in $\mathscr L^{2}$, then inequality holds with equality

From Williams' Probability with Martingales I tried rewriting the RHS to: $$\sum_{k=n+1}^{\infty} E[(M_k - M_{k-1})^2] = \sum_{k=n+1}^{n+r} E[(M_k - M_{k-1})^2] + \sum_{k=n+r+1}^{\infty} ...
0
votes
1answer
27 views

Prove $A_{\infty} < \infty$?

From Williams' Probability with Martingales How do we know that $A_{\infty} < \infty$? If $T = \infty$, then $$E[A_{T \wedge n}] \le (K+c)^2$$ $$\to E[A_{n}] \le (K+c)^2$$ $$\to ...
-1
votes
0answers
49 views

$X_n$ doesn't converge to a limit in $[-\infty, \infty] \to$ Is this supposed to be a stronger version of $\lim X_n$ doesn't exist?

From Williams' Probability with Martingales: What's the difference between saying that '$X_n(\omega)$ does not converge to a limit in $[-\infty,\infty]$' and '$\lim X_n$ does not exist' ? ...
0
votes
3answers
53 views

Finding a Taylor Series Representation: $f(x)= \frac{x}{(1+4x^2)^2}$ [closed]

Find Taylor series representations for the following function. For precisely what values of $x$ is the series representation valid? $$f(x) = \frac{x}{(1+4x^2)^2}$$
3
votes
1answer
34 views

Definition of convergence of a sequence

Can this be a valid definition? For each $\epsilon>0$, there exists $K \in \Bbb N$ such that if $n\ge K$, then $|a_n-a|\le\epsilon$. Should I use "$<\epsilon$" or "$\le \epsilon$"?
3
votes
3answers
32 views

A different notion of convergence for this sequence?

I was thinking about sequences, and my mind came to one defined like this: -1, 1, -1, 1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, ... Where the first term is -1, and after the nth occurrence of -1 in the ...
-1
votes
2answers
49 views

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? [duplicate]

Let $f_n(x)=x-x^n$ for $x \in [0,1]$ Does the sequence ${f_n}$ converge pointwise on the set $[0, 1]$? This is what I have done, since $0\le x\le 1$ $ x=0$$$f_n(0)=0-0^n=0$$ $x\to \infty$ ...
1
vote
0answers
29 views

Convergence of $\sum \frac{a_n}{1+a_n}$ implies convergence of $\sum a_n$ for positive $a_n$. [duplicate]

I need to prove or disprove the statement. I think the statement is true. My attempt at a proof: From the definition of convergence: $$\forall \epsilon > 0 \quad \exists N \in \mathbb{N} \quad ...
0
votes
0answers
16 views

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$

Show that the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \cdot y_k \rvert$ converges if the absolute infinite series $\sum_{k=1}^{\infty} \lvert x_k \rvert$ converges too and the ...
2
votes
3answers
87 views

Does the infinite series $\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$ converge?

I have been wondering if this infinite series converges $$\sum_{k=1}^{\infty} \frac{1}{(4+(-1)^k)^k}$$ I tried to put it in wolfram alpha but it says that the ratio test is inconclusive, but when I ...