Convergence of sequences and different modes of convergence.

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14 views

Convergence to $N(0,1)$ in distribution

Question is as following: $X\in Po(\lambda)$ $\frac{X-\lambda}{\sqrt{\lambda}} \,{\buildrel d \over \rightarrow}\, > N(0,1)$ as $\lambda \rightarrow \infty$ I begin by setting ...
3
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1answer
907 views

Why is the ratio test for $L=1$ inconclusive?

One of the often used tests for convergence ($L\lt 1$) and divergence ($L\gt 1$) of an infinite series is the ratio test. The idea behind it, why it works is the geometric series which dominates (or ...
-1
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1answer
79 views

What's the limit of this sum $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ [on hold]

Let $ S_n=1^s+\frac{1}{2^s}+3^s+ \frac{1}{4^s}+5^s+\frac{1}{6^s}+\cdots+n^s+\frac{1}{(n+1)^s} $ and s be a complex variable . $s=\sigma +it $ where $\sigma ,t \in\mathbb{R} $ , Note :I edit the ...
9
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3answers
96 views

A series involving $\prod_1^n k^k$

Is this series $$\sum_{n\geq 1}\left(\prod_{k=1}^{n}k^k\right)^{\!-\frac{4}{n^2}} $$ convergent or divergent? My attempt was to use the comparison test, but I'm stuck at finding the behaviour of ...
0
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1answer
30 views

$\mathbb E[\bar X_n]=0$

A conditional normal rv sequence, does the mean converges in probability, in this question how can i get $\mathbb E[\bar X_n]=0$? Here is my attempt; $$\mathbb E[\bar ...
1
vote
1answer
37 views

Applying the dominated convergence theorem to $\lim_{n\to\infty} x^n$, for $x \in [0,1]$.

$\lim\limits_{n\to\infty} x^n$, for $x \in [0,1]$. I'm using the dominated convergence theorem on a few problems and keep running into this issue. What's the limit of the above function? ...
2
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2answers
63 views

Computing the limit of an alternating series,

I am looking at the series $$ \sum_{n=1}^\infty\frac{(-1)^n}{n}.$$ This series converges (conditionally) by the alternating series test. How can I compute its limit, which is equal to -log(2)? a) ...
5
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2answers
54 views

Convergence of series of $1/n^x$ - pointwise and uniformly,

Consider the series $$\zeta(x) = \sum_{n\ge 1}\frac {1}{n^x}.$$ For which $x \in[0,\infty)$ does it converge pointwise? On which intervals of $[0,\infty)$ does it converge uniformly? My work: I ...
1
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1answer
33 views

$G_n:=\sqrt{n} \left(X_n-1\right) \xrightarrow[n]{d} N(\mu,\sigma^2) $ implies $\sqrt{n} \left(1-X_n^{-1}\right)=G_n+o_P(1)$

Let $X_n$ be a sequence of RV so that $G_n:=\sqrt{n} \left(X_n-1\right) \underset{n \to \infty}{\overset{d}{\longrightarrow}} G \sim N(\mu,\sigma^2)$. I want to show that in this case $\sqrt{n} ...
1
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1answer
23 views

$L^p$ Martingale convergence theorem

I am trying to prove the $L^p$ Martingale convergence theorem for martingale $X=(X_n)^{\infty}_{n=0}$ on $(\Omega,\mathcal{F},(\mathcal{F}_n)^\infty_{n=0},\mathbb{P})$ which is bounded in $L^p$ for ...
2
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2answers
34 views

Finding the limit of this integral: $\lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$ if $q<p+1$

I am trying to find the following limit provided: $q<p+1$: $$ \lim_{n\to\infty} \int_0^1 \dfrac{n x^p+x^q}{x^p+n x^q} dx$$ Dividing by $n x^q$ so we have $$\dfrac{n x^p+x^q}{x^p+n ...
4
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2answers
353 views

If $\,x>1$, then $\displaystyle\lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x$.

How can I prove that $$ \lim_{n\rightarrow\infty}\frac{\left\lfloor x^{n+1} \right\rfloor}{\left\lfloor x^n \right\rfloor}=x, $$ whenever $x>1$. Here $\left\lfloor \cdot\right\rfloor$ denotes the ...
7
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1answer
245 views

Compute limit of the sequence $x_n$ given by $x_{n+2}=-\frac{1}{2}(x_{n+1}-x_n^2)^2+x_n^4$

Let $(x_n)$ be a real sequence such that $x_0=a\in\mathbb{R},x_1=b\in\mathbb{R},x_{n+2}=-\dfrac{1}{2}\left(x_{n+1}-x_{n}^2\right)^2+x_{n}^4\;\forall n\in\mathbb{N} $ and $|x_n|\leq ...
2
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2answers
47 views

If $\{x_n\}$ is a sequence in $\mathbb{N}$ and $x_n \rightarrow x$, prove there exists $N$ such that $x_n = x$ for $n \geq N$

Since $x_n \rightarrow x$, we know that for all $\epsilon > 0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$, $|x_n - x| < \epsilon$. We want to show that for some $\epsilon ...
3
votes
1answer
56 views

Writing a proof of the convergence of a series defined recursively

Define the sequence $a_n$ recursively by $a_1=1$ and $$a_{n+1}=\frac13\left(a_n^2+\frac1n\right)$$ (a) Prove, by induction or otherwise, that $(a_n)$ is decreasing. (b) Prove that the series ...
4
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0answers
28 views

A conditional normal rv sequence, does the mean converges in probability

$X_1, X_2, \dots, X_n$, are $n$ mutually independent r.v.s. $Y_1,\dots,Y_n$ are another set of mutually independent r.v.s. $X_k\mid Y_k=y_k\sim N(y_k,y_k^2)$ and $Y_k\sim\text{uniform}(-k,k)$ for ...
3
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1answer
50 views

Writing solution to an arbitrary ODE with arbitrary initial values as the sum of a power series?

How can we solve for $y$ with these arbitrary initial values and polynomials? How would we write the solution as a power series?
4
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1answer
37 views

Almost surely vs expectation

Let $X_1, X_2, X_3 \dots$ be a sequence of random variables. In the limit as $i \rightarrow \infty$ we have $$ X_i \rightarrow 0 \text{ almost surely} $$ Does it follow that In the limit ...
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1answer
53 views

A probability problem with maximum and summation

Let $X_n$ be iid nonnegative r.v.s, suppose there exists positive sequence $a_n$ such that $S_n/a_n\xrightarrow[]{P}1$, then show $$\max_{1\le i\le n} X_i/S_n\xrightarrow[]{P}0.$$ I have shown that ...
2
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1answer
160 views

If $\int(f_n) \rightarrow \int(f)$ then $\int(|f_n-f|) \rightarrow 0$ for $f_n \rightarrow f$ pointwise

I'd like to show that for an integrable sequence of functions $f_n:X \rightarrow [0, \infty)$ with $\sup_{n\geq 1} \int_{X} f_n d\mu < \infty, f_n \rightarrow f$ pointwise a.e. for some function ...
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2answers
48 views

Does alternating test show divergence?

My book states the alternating tests' convergence requirements. However, my book doesnt point out, if $a_n$ fails one of the convergence requirements, is it true that is diverges? Such as the limit ...
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1answer
36 views

Interval of converge of $\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$

Find the interval of converge of: $$\sum_{n=1}^{\infty} \frac{n!(x+1)^n}{(2n-1)!}$$ I will use the ratio test. Let $\displaystyle a_n = \frac{n!(x+1)^n}{(2n-1)!}$ $\displaystyle a_{n+1} = ...
1
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1answer
5k views

How to determine if the series $\sum \frac{2+\sin n}{5^n}$ is convergent or divergent?

I have: the series $\sum_{n=0}^{\infty} \frac{(2+\sin n)}{5^n} $. I have split this up into $\sum_{n=0}^{\infty} \frac{2}{5^n} + \sum_{n=0}^{\infty} \frac{\sin n}{5^n}$. I know the first part is ...
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1answer
13 views

Convergence in distribution of the negative part of centered/scaled poisson variable

For every real number $x$ denote its negative part by $x^{-}$ if $x \le 0$, and let $x^{-} = -x$. Otherwise let $x^{-} = 0$. Now let $$T_n = \frac{(X_1 + \ldots + X_n) - n}{\sqrt{n}}$$ where $X_j ...
0
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0answers
30 views

I need to find a rational numbers series that converging to irrational number [duplicate]

I found a series that is $a_{n+1}=\frac{a_n^2 + 2}{2a_n}$ yet I'm not sure. can someone give me a more umm solid example? thanks.
23
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2answers
854 views

Find the exact value of the infinite sum $\sum_{n=1}^\infty \{\mathrm{e}-(1+\frac1n)^{n}\}$

How can we find the exact value of the infinite sum $$ \displaystyle\sum_{n=1}^\infty \left\{\mathrm{e}-\Big(1+\frac1n\Big)^n\right\}? $$ This problem appears in: T. Andreescu, T. Radulescu & V. ...
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0answers
50 views

How to prove that the series converges?

Let us suppose that $\vert a_n\vert$ is a real sequence, and set $S_n=\sum_{k=1}^{n}a_k$ and $\sigma_n=\frac{1}{n+1}\sum_{k=1}^{n}S_k$. How can we show that if the series $\sum_{n=1}^{\infty}\vert ...
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0answers
8 views

Convergence of third moment in central limit theorem

Previously, I asked a question here about the rate of convergence of expectations of absolute values to the expected value of a Gaussian. If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = ...
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2answers
36 views

Convergence of sets implies containment [on hold]

Let l$_1$ be a set of {x$_n$} such that $\Sigma$$|x_n|$ is convergent. Let l$_2$ be a set of {x$_n$} such that $\Sigma$$|x_n|$$^2$ is convergent. So: l$_1$= {{x$_n$}$\in$R: $\Sigma$$|x_n|$ l$_2$= ...
2
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1answer
41 views

Prove $f:\mathbb{N} \rightarrow \mathbb{R}$ is continuous using the definition of sequential continuity

The definition of sequential continuity is that $x_n \rightarrow x \implies f(x_n) \rightarrow f(x)$. If the terms of the sequence $\{x_n\}$ are only natural numbers, I know that for all $\epsilon ...
0
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1answer
28 views

Power Series Solutions And Minimum Radius of convergence [on hold]

Help with power series and minimum radius of convergence. Does the equation $$ (x^2 + 25)y'' + xy' + x^3y = 0 $$ have a power series solution $y = \sum_{n=0}^\infty c_n x^n$? If yes, ...
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1answer
39 views

Can someone help me prove that this sequence converges? [on hold]

I'm having difficulty trying to prove this, mostly because I don't understand the process. There was a proof similar to this in my textbook where they proved if Sn converges to s then lim 1/Sn=1/s and ...
0
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1answer
18 views

Special case of limsup inequality

If you are given two sequences $a_{n}$ and $b_{n}$ such that $a_{n}b_{n} \geq 0$ and the limits $\lim\limits_{n \rightarrow \infty}a_{n} = a$ and $\lim\limits_{b \rightarrow \infty}b_{n} = b$ then can ...
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1answer
19 views

$L_1$ convergence of $\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$

Does the sequence $f_n=\frac{1}{\sqrt{x}}\sin{\left(\frac{1}{nx}\right)}$ on $(0,1)$ converge in $L_1$? It converges to zero pointwise and I think it converges in $L_1$ as well since ...
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1answer
39 views

What is the solution to poissons equation with a point source?

I am currently trying to solve the following PDE using various numerical methods - direct and iterative. $$ \nabla ^2 u = \rho $$ where $u = u(x,y)$ and $\rho(0.5,0.5) = 2$ (zero elsewhere). The ...
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1answer
51 views

Problems on sequence and series of functions

Let $a_n$ be a sequence of real numbers. Which of the following is true? a. If $\sum a_n$ converges,then so does $\sum a_n ^4.$ b.If $\sum |a_n|$ converges,then so does $\sum a_n ^2.$ ...
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1answer
45 views

Prove or disprove convergence in distribution of a poisson variable.

Let $$S \overset{d}{\sim} Poisson(\lambda).$$ I would like to determine $\frac{S-\lambda}{\sqrt{\lambda}}$ converges in distribution as $\lambda \rightarrow \infty.$ So my set up is: $$\Pr\left[a ...
0
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1answer
34 views

What is the difference between the limit of a sequence and a limit point of a set?

I always thought they were the same thing. The limit of a sequence is a point such that every neighborhood around it contains infinitely many terms of the sequence. The limit point of a set is a point ...
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2answers
75 views

Don't understand proof that $x_n \rightarrow A$ $\iff$ every subsequence of $\{x_n\}$ converges to $A$

So we are given that for all $\epsilon > 0$, there exists an $N$ such that for all $n \geq N$, we have that $|x_n - A| < \epsilon$ and we want to show that for all $\epsilon' > 0$, there ...
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3answers
32 views

Can someone help me understand the proof that every cauchy sequence is bounded?

This proof is written by a user Batman as an answer to someone's question(just to give credit). Every proof that I've seen is the same idea, and I'm having trouble understanding it intuitively. (I ...
2
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1answer
85 views

A step in proving that a real Cauchy sequence is convergent.

I'm trying to prove that a real Cauchy sequence is convergent, but I need some help for a step. We have the following statements: $\{ s_i\}$ is a real Cauchy sequence, i.e. ...
2
votes
1answer
81 views

Prove $\lim_{n\to\infty}[(n^2+n)^{1/2}-n]= 1/2$

So I know that this for all epsilon>0, there exists an N such that for all n>N dp$ (p_n,\frac{1}{2})<epsilon $. But in this particular example I'm having difficulty finding the $N$ for which this ...
5
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5answers
131 views

Prove that $c_n = \frac1n \left(\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}} \right)$ converges

I want to show that $c_n$ converges to a value $L$ where: $$c_n = \frac{\large \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}}{n}$$ First, it's obvious that $c_n > 0$. I ...
0
votes
1answer
41 views

Does the sequence $( n^{1/n} -1)$ belong to any $\ell^p$ space? [on hold]

The sequence $( n^{1/n} -1)$ converges to zero but does this sequence belong to the $\ell^p$ space for $p\in\mathbb R$? I don't know the answer, or how one would prove it. Same problem with the ...
2
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2answers
54 views

how to find convergence and divergence of the series [closed]

consider the following two series of complex numbers $$s_1=\sum_1^\infty\frac{i^{n}(2-\sin n)}{2^n.n}$$ $$s_2=\sum_1^\infty\frac{i^n(2-\sin n)}{2^n.n^2}$$ then a)both the series converge ...
6
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6answers
130 views

Convergence of $\sum_{n=1}^{\infty}\left(\, \frac{1}{n} - \frac{1}{n + 2}\,\right)$

What criteria can I use to prove the convergence of $$ \sum_{n=1}^{\infty}\left(\,{1 \over n} - {1 \over n + 2}\,\right)\ {\large ?} $$ My idea was to use ratio test: $$\displaystyle{1 \over n} - {1 ...
0
votes
1answer
17 views

uniform convergence of diff'ble functions with derivatives converging in $L^1$

Suppose we have a sequence of differentiable functions $f_n$ from a closed interval $I\subseteq\mathbb{R}$ to $\mathbb{R}$ with the following properties: $f_n$ converges uniformly to some function ...
0
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0answers
18 views

Correct definition for convergence of a subsequence?

I only have the definition for convergence of a sequence, but can't find a definition for convergence of a subsequence. I have two guesses: For all $\epsilon > 0$, there exists an $N \in ...
4
votes
1answer
72 views

Convergence in measure implies convergence almost everywhere (on a countable set!)

Here is an interesting problem from "Real Analysis for Graduate Students" by Richard Bass (which is an amazing book, by the way). Suppose $(X, \mathcal{A}, \mu)$ is a measure space, and $X$ is a ...
3
votes
2answers
67 views

If $X_i$ are iid $U(0,1)$ random variables, $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$

I want to show $\max_{1\le i \le \frac{n}{2}}\{(1-\frac{2i}{n})X_i\}$ converges in probability to $1$ as $n \to \infty$, where $X_i$ is an i.i.d sequence of $[0,1]$-uniformly distributed random ...