Convergence of sequences and different modes of convergence.

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2
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Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences

Here're Definitions 3.15 and 3.16 and Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Definition 3.15: Let $\{s_n \}$ be a sequence of real numbers ...
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1answer
47 views

Prove $\sum_{n = 0}^{\infty} a_n X^n$ converges at every point in $[-1,1)$ if $a_n$ is non-increasing and converges to $0$

Prove that if $a_n$ is non-increasing, converges to $0$, and radius of convergence of $\sum_{n = 0}^{\infty} a_n X^n$ is equal to $1$ then this series converges at every point in $[-1,1)$ I am trying ...
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3answers
33 views

Proof about infinite limsups and subsequences

I am working on a final exam study guide, and came across this question: Suppose limsup$(a_n)$ = $\infty$. Prove: There must exist a sub-sequence ${a_n}_k$ such that ${a_n}_k \to \infty$. My initial ...
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2answers
48 views

Metric space and continuity

We define a map $f:(S,d)→(S',d')$ between 2 metric spaces to be continuous at x belongs to S if for every sequence ${x_n}$ in $S$ that converges to x, the sequence {f(x_n)} in $S'$ is convergent to ...
2
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2answers
28 views

Difficulty finding a power series representation

I have to find a power series representation and interval of convergence for $$f(x) = \frac{x-x^2}{(1+2x)^3}$$ Noting that $\frac{1}{1+2x}=\frac{1}{1-(-2x)}=\sum_{n=0}^\infty(-2x)^n$, I start taking ...
2
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0answers
35 views
+50

Convergence of moments of a sequence of random variables

I encountered this problem in my study of time series. It seemed trivial at first but I don't see the finishing move to complete the proof. The problem is as follows. Let $(X_n)_n$ be a sequence of ...
2
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2answers
52 views

Show : $\displaystyle \sum_{n=1}^\infty \dfrac{a_n}{1+a_n}$ is absolutely convergent [duplicate]

Given $\displaystyle \sum_{n=1}^\infty a_n$ is absolutely convergent. Show that $\displaystyle \sum_{n=1}^\infty \dfrac{a_n}{1+a_n}$ also converges absolutely. (If $a_n \neq -1, \forall n \geq 1$ ) I ...
0
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2answers
42 views

Expand $f(x)=\log(x+ \sqrt{1+x^2})$ into power series and determine convergence intervals

We have $f(x)=\log(x+ \sqrt{1+x^2})$ and we need to expand it into power series, which I suppose is easy because $f'(x)=(1+x^2)^{-1/2}= \sum_{k=0} {-\frac{1}{2} \choose k}x^{2k}$. It follows that ...
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0answers
23 views

Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to $f^{ (k)}$ uniformly on any compact subset of $G$.

Suppose $f_n$ is analytic in some region $G$ and suppose $f_n$ converges to $f$ uniformly on any compact subset of $G$. Prove that $f$ is analytic and derivatives of all order $f_n^{(k)}$ converge to ...
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3answers
80 views

Why $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over (n^2) }$ converges?

I have to prove that the series $\displaystyle\sum_{n=1}^\infty {\ln(n^2) \over n^2 }$ converges. The ratio test is inconclusive, so I should use the comparison test, but which series should I compare ...
0
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0answers
12 views

Equality involving holomorphic function and its series coefficients [duplicate]

Function $f(z)=a_0 + a_1z +a_2z^2+...$ convergences on $\left\{z:|z|<R\right\}$. Prove that for any $0<r<R$ $$\frac{1}{2\pi}\int_{0}^{2\pi}|f(re^{it})|^{2}dt= ...
0
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0answers
24 views

About accumulation point in compact metric space

Let $(X, d)$ be a compact metric space, and $\{x_n\}_{n\in N}$, $\{y_{n,m}\}_{m,n\in N}$ be subsets of $X$. Question: Is there a subsequence $\{n_k\}$ such that $x_{n_k}\rightarrow x$ and ...
0
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4answers
32 views

Is there convergence in this sequence?

I have the sequence $a_{n} = \frac{1}{5n^2 + \cos(n\pi )+1}$ $n\in \mathbb{N}$ It's obvious that it converges to 0 but I have problems to proof it: Let $\varepsilon$ be optional and choose $N$ as ...
0
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0answers
20 views

On a closed-form from particular values of the Riemann zeta function and divisor functions

I am looking if I can get a closed-form for an infinite series, but I don't know for what it is possible, without finish my computations (see my Question, below). From Applications (8.1 Infinite ...
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1answer
38 views

Spread of a rumor in a growing population

This is a variation on a classic problem. It occur's in several problems I am researching and I'd like to get some help from folks who may have dealt with this already or can offer insights. Let ...
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3answers
27 views

evaluate the following limit else prove that limit does not exist

The function/sequence of interest is as follows: $(\frac{n!}{n!+2})^{n!}$ I have a feeling the limit does exist, as if we divide the numerator and denominator by $n!$ we get ...
1
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1answer
27 views

Why is $\overline{B(l^2)\odot B(l^2)}^{\| \enspace \|_{op}}\neq B(l^2\otimes l^2)?$

Let $B(l^2)$ be the $C^*$algebra of bounded linear operators on the sequence space $l^2$ and denote with $B(l^2)\odot B(l^2)$ the tensor product of $B(l^2)$ with itself, considered as a $*$algebra ...
0
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0answers
23 views

On the zeroes of sine complex function, and a search for a special sequence, following Riemann's approach

If there are no mistakes from the Fourier expansion series for the fractional part function we can write, using a substituion, that for $1<x<e^2$ with uniform convergence $$\frac{1}{2}\log ...
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1answer
42 views

Proof a series is coverges to a specific sum

I need to prove that the sum of the following series is convergent to : $1 \ge Sum$ $$\sum_{n=1}^\infty \ ...
0
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0answers
14 views

Prove that the following function is in $H^1(\Omega)$

Let $\Omega$ be such that $$ \overline{\bigcup_{k=1}^\infty \{b_k\}}^{|\cdot|} =\Omega:=\left\{(x_1,x_2)\in\mathbb R^2; \sqrt{x_1^2+x_2^2}<1/2\right\}, $$ where $|\cdot|$ denotes the Euclidean norm ...
1
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1answer
24 views

Convergence of a series implies convergence of another series

Let $a_1,a_2,\cdots$ be a sequence of real numbers with $a_i\geq 0$. If $\sum_{n=1}^{\infty}\frac{1}{1+a_n}<\infty$ then show that $\sum_{n=1}^{\infty}\frac{1}{1+x_na_n}<\infty$ for each real ...
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2answers
63 views

Convergence of $\sum _{n=3} ^\infty \frac 1 {(\ln \ln n)^{\ln \ln n}}$

$$\sum _{n=3} ^\infty \frac 1 {(\ln \ln n)^{\ln \ln n}} .$$ I believe the series diverges. I am thinking of using the integral test to show this, but I am not sure if that is right.
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2answers
37 views

Converging or Diverging Series

What test do i use to show this series converges or diverges? $$\sum_{r=1}^{\infty}\frac{1}{(1+\frac{1}{r})^{r}}$$ I know that $(1+\frac{1}{r})^{r} \rightarrow e$ so does this function converge to ...
1
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1answer
28 views

Can a discrete function converge to a continous function?

Let $f\in C^{\infty}[a,b]$, let also $X \subset [a,b] = \left\{x_0,\ldots,x_k \right\}, Y = \left\{ f(x_0),\ldots, f(x_k) \right\}$. I guess that if I let $k\rightarrow \infty$ then some how I should ...
0
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1answer
25 views

If $\{f_n\}$ is Cauchy in measure, then there is a measurable function $f$, such that $\{f_n\}$ converges in measure to $f$

The theorem is from Real Analysis (Carothers). Let $\{f_n\}$ be a sequence of real valued measurable functions, all defined on a common measurable domain $D$. If $\{f_n\}$ is Cauchy in measure, then ...
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18 views

Is this sum with fractional part function and harmonic numbers convergent?

About an hour ago I went to bed and the following question came, so I got up from the bed in order to share it with you, and because I would like to see the solution. I apologize if this is something ...
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3answers
51 views

Showing that $\int_0^{+\infty} \frac{\sin(1/x)}{x}\ dx$ converges

How can I show that $\int_0^\infty \frac{\sin(1/x)}{x}\ dx$ converges? I have that $\sin(x)\leq x$ for $x\geq 0$ so then $\sin(1/x)\leq 1/x$ for $x\geq 0$. It follows then that $\int_1^\infty ...
0
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1answer
26 views

Radius of convergence of two series [duplicate]

An unproven proposition in my book states that if the series of $a_{n}z^n$ has radius of convergence $R_1$ and the series of $b_{n}z^n$ has radius $R_2$. Then the radius of convergence of ...
-1
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2answers
31 views

Find all values of x for which the infinite series $S = \sum_{n=0}^\infty\left(\frac{x^2}{x^2+1}\right)^n $ converges

Find all values of x for which the infinite series $S = \sum_{n=0}^\infty\left(\frac{x^2}{x^2+1}\right)^n $ converges, and express $S$ as a function of $x$. I think the interval of convergence is ...
0
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0answers
27 views

How to prove this continuous martingale converges?

Suppose $B = (B_t, t \geq 0)$ is standard Brownian motion. Let $M^\lambda_t := \exp(\lambda B_t - \frac{\lambda^2 t}{2})$ (I have previously shown that this is a martingale). How do I prove that $$ ...
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3answers
64 views

Show whether this trigonometry series converges

$\displaystyle\sum_{n=1}^{\infty}\sin\left(\frac{3n}{1+3^n}\right)$ Kinda obvious that it converges, but how do I prove it mathematically?
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1answer
15 views

Equivalents definition of linear convergence

Suppose that the sequence $\{x_n\}$ converges to $0$. I want to prove that these definitions are equivalent: a) We say that $\{x_n\}$ converges linearly to $0$, if there exists a number $q \in (0, ...
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2answers
46 views

Definition of limit convergence

I want to show that the sequence represented by .9, .99, .999, ... converges to 1 but I'm unsure how to go about picking an epsilon such that it works
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2answers
118 views

If $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = 0$, can $\sum_0^\infty a_n$ be rational?

If a nonzero sequence of rationals $$a_0, a_1 \dots a_n$$ "decays fast" in the sense that $\lim_{n \rightarrow \infty} a_{n+1}/a_n = 0$, can the series converge to a rational number? That is, can ...
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0answers
21 views

Examine convergence and (almost) uniform convergence of $f _{n} = n\left[ f\left( x + \frac{1}{n} \right) - f\left( x\right) \right] $

Show almost uniform confergence of: $ f _{n} = n\left[ f\left( x + \frac{1}{n} \right) - f\left( x\right) \right] $ I've noticed that : $f' _{n} = \frac{ f\left( x + \frac{1}{n} \right) - ...
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2answers
50 views

Show that a power series is analytic inside its radius of convergence

Let $f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k$ with radius of convergence $R$ then $f$ is analytic on the open disk around $z_0$ with radius $R$. What I was thinking about is an approach based on this ...
6
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2answers
48 views

Nested logarithm series

While working on problems from Spivak's Calculus, I came on one asking for the convergence/divergence of the series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}.$$ This is a straightforward ...
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0answers
21 views

Radius of convergence of $f(z)?$

Let $p(x)$ be a polynomial of the real variable $x$ of degree $k\geq 1$ .Consider the power series $$f(z)=\sum_{n=0}^{\infty}p(n)z^n$$ where z is a complex variable .Then the radius of convergence of ...
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1answer
14 views

ratio test is inconclusive

I have a series $\sum_{k=1}^{\infty} \frac{cos(k)}{2*(k^3)-k}$ Using ratio test, I got $={\sum_{k=1}^{\infty} \frac{cos(k+1)}{k*(2*(k+1)^2-1)}}/{\frac{cos(k)}{k*(2*(k^2)-1)}} = ...
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0answers
29 views

Check convergence $ \sum_{}^{} \frac{n ^{2}x ^{2} }{e ^{n ^{2}\left| x\right| } } $

Check the pointwise, uniform, and almost uniform convergence of: $ \sum_{}^{} \frac{n ^{2}x ^{2} }{e ^{n ^{2}\left| x\right| } } $
3
votes
4answers
89 views

Convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$

I want to test the convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$. There are some parts of the solution which does not make sense to me, I'm hoping that someone can explain ...
0
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1answer
52 views

Examine convergence and (almost) uniform convergence of $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$.

How to examine convergence, almost uniform convergence and uniform convergence of series $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$?
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13 views

Monte Carlo with non uniform weighting

So, I just want to check if what is in my mind is in fact true. Assume, that we have a distribution over the whole $\mathbb{Z}^+$, where $p(k) = \gamma_k$. We are interested in approximating $p(v)$ ...
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0answers
7 views

An examination of rates of convergence of the series

I check the website they all talked about rate of convergence of sequence. Can anyone gives me an example for rates of convergence of the series?
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1answer
18 views

Sum of two random variables converging with different modes [closed]

Is it true that if X_n converges in distribution to X; Y_n converges in probability to Y; X_n, Y_n, X and Y are real-valued random variables defined on the same probability space, then X_n + Y_n ...
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votes
3answers
47 views

Determine whether the series converges

$\sum\limits_{n=1}^{\infty} \dfrac{1}{n^{1+1/n}} $ I believe it diverges. But I am having trouble comparing it to another series which also diverges and whose terms are less than the original ...
0
votes
1answer
47 views

Given that an integral containing $|x_{n}|$ is bounded by a constant, show that $x_{n}\to x$ in $L^{1}(0,\infty)$

This question is related to another question I asked earlier. For reference, this is the relevant part of that question: Let the sequence of continuous functions $\mathbf{\{x_{n}(t) ...
-2
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0answers
23 views

explain why there is an observed rate of convergence

Using your knowledge and theorems explain why there is an observed rate of convergence when using the composite simpsons rule?
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0answers
20 views

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$ and prove that $B_n=\{m \in \mathbb N : d(m,n)\leq ...
0
votes
1answer
17 views

Criteria for convergence of power series

Given the power series: $\; \sum_{i=0}^{\infty}a_nz^n \;$ Proof that if there exist $s,M \in \mathbb R $ such that $|a_n|s^n \le M$ then the power series converges for every $|z|\lt s$ Can someone ...