Convergence of sequences and different modes of convergence.

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2
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2answers
71 views

$L_p$ space and convergence

Let $f_i\rightarrow f$ $m$-a.e on $[0,1]$, $m$ is a measure and $\int|f_i(x)|^4dm$$\le1$ for all $i$.Then $\int|f_i(x)|^2dm\rightarrow \int|f(x)|^2dm$. how to prove it? in my solution i prove that ...
0
votes
0answers
29 views

Infinite sequences with all but finitely many elements equal

Consider two infinite sequences $\{a_1,a_2,\dots\}$ and $\{b_1,b_2,\dots\}$ such that $0<|a_i|<1$, $0<|b_i|<1$, $|a_i|\geq|b_i|$ for all $i$ and $a_i=b_i$ for all except finitely many $i$. ...
0
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0answers
27 views

Convergence/Divergence relation to Derivative

I am a student who just finished Calculus BC in high school (equivalent to Calculus II in college). Firstly, the definition of convergence I was presented with by my high school teacher was that ...
0
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2answers
15 views

Convergence test from Demidovich

I'm just learning how to test series for convergence and have encountered this series from the Demidovich's book and I can't really decide what criteria should I use. Could you please give me some ...
3
votes
1answer
13 views

Product of weak and strong convergent sequences in $L^p$

I already saw some proofs here with $b_n\to b$ in $L^2$ and $a_n\rightharpoonup a$ in $L^2$. Then $$ \int a_n b_n \to \int a b. $$ But what goes wrong if both sequences are weak convergent? Proof: ...
0
votes
0answers
57 views

convergence of series $\sum_{n=2}^\infty\frac{\sin(\frac{1}{n})}{\ln(n)}$

I am unable to test convergence for $$\sum_{n=2}^\infty\frac{\sin(\frac{1}{n})}{\ln(n)}\;,\; \sum_{n=2}^\infty\frac{\sin(\frac{1}{n})}{\ln^2(n)}$$ I know from wolfram that the first diverges and ...
0
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1answer
43 views

Prove regarding net convergence

Let $M$ be a non-empty set and let $\mathfrak{X}$ be the set of all finite subsets of $M$. Let $\mathfrak{X}$ be a directed set by the relation $$\forall X,Y \in \mathfrak{X} \:\:\: X\le Y ...
-1
votes
2answers
72 views

$\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$?

Is the following true or false: If $\int_0^{\infty} f(x)dx$ converges and $\lim_{x\rightarrow \infty} f(x)$ exists, then $\lim_{x\rightarrow \infty} f(x)=0$? This should be doable without series.
1
vote
0answers
48 views

If $X_n\to0$ in probability then $X_n/r_n\to0$ in probability, for some $r_n\to0$

Suppose that a sequence of positive random variables $X_n$ is such that for all $\epsilon >0$, $P(X_n>\epsilon)\rightarrow 0$ as $n\rightarrow\infty$, that is $X_n$ converges in probability ...
2
votes
1answer
36 views

Can you provide us a good approximation for $\sum_{n=1}^{\infty} \left| \log \left( 1+\frac{\mu(n)}{n^2} \right) \right|$?

Let $a_n=\frac{\mu(n)}{n^2}$, where $\mu(n)$ is the Möbius function. Since $\sum \left| a_n \right| $ is convergent by the comparison test, then a proposition from analysis ensures that ...
8
votes
1answer
63 views

Is it possible to study the properties of sequences by studying the family of polynomials generated with the elements as coefficients?

Suppose there is an integer sequence $\{a_0,a_1...a_n...\}$ and a family of polynomials is defined as follows: $p_0 = a_0$ $p_1 = a_0x+a_1$ $p_2 = a_0x^2+a_1x+a_2$ $p_n = ...
0
votes
0answers
25 views

Calculate the radius of convergence of the following power series

Let the power serie $\sum_{k\ge0}a_k(z-a)^k$ have the radius of convergence $\rho=t\in\mathbb{R^+}$, and let $p\in\mathbb{N}$. What is the radius of the following series: a) ...
1
vote
1answer
71 views

What is the convergence radius of $\sum_{n=0}^\infty a_nx^n$ when $\{a_{n}\}$ is s.t. $a_1 = 1, a_{n+1} = \sin(a_{n})$?

My task is this: Given a sequence $\{a_n\}$ with $a_1 = 1, a_{n+1} = \sin(a_{n}).$ (i) Show that the sequence converge and find the limit as $n\to\infty$. (ii) Show that $\sum_{n=0}^\infty a_nx^n$ ...
1
vote
2answers
71 views

Convergence/divergence of a messy integral: $\int_1^\infty\frac{\arctan(9x)}{1+9x^3}\:dx$

Considering $$ \int_1^\infty\frac{\arctan(9x)}{1+9x^3}\:dx $$ I am trying to show convergence but looking to use Dirichlet's test and wanted to see if we can do it this way. Are we supposed to show ...
2
votes
1answer
53 views

How to prove the limit $\lim\limits_{n\to\infty}n\sin\frac{2\pi}{n}\cos\frac{1}{n}$ doesn't exist?

I am at a loss... how do I prove that the limit $\lim_{n\to\infty}(n\sin\frac{2\pi}{n}\cos\frac{1}{n})$ doesn't exist?
2
votes
1answer
53 views

Does $\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n$ converge at the endpoints of the convergence radius?

My task is this: Find the convergence radius of$$\sum_{n=0}^\infty \frac{(n!)^2}{(2n)!}x^n.$$ My work so far: By ratio test we get ...
-2
votes
2answers
44 views

Proof that both series $\sum a_n$ and $\sum b_n$ are either convergent or divergent [closed]

Proof, that if $c_1a_n< b_n < c_2a_n$ with $c_1>0$ , $c_2>0$ , sequence $a_n>0$ and $b_n>0$ with for all $n>n_0$ and $n_0$ is a natural number, that both series $\sum a_n$ and ...
0
votes
1answer
15 views

Almost sure convergence of the inverse

If a sequence of non-negative random variables $X_1, X_2, \dots$ converges almost surely to a random variable $X$, that is $X_n \xrightarrow{a.s} X$ or equivalently ...
1
vote
0answers
25 views

Convergence in distribution for two random variables

If $\lim\limits_{n \to \infty} P(X_n\leq T)=P(X\leq T)$ and $\lim\limits_{n \to \infty} P(Y_n\leq T)=P(Y\leq T)$, where $X_1, X_2,\cdots$ and $Y_1, Y_2,\cdots$ are two sequences of random ...
0
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2answers
29 views

Proving a sequence defined by a recurrence relation converges

How can I prove that this iterative sequence converges to $2$? Can I use the convergence definition? $$a_{n+1} = \frac{a_n}{2} + \frac{2}{a_n},\qquad a_0 = 4 $$ Thanks for the help.
2
votes
1answer
20 views

Almost sure convergence of a function and its argument

If $f_n(x)\xrightarrow{a.s}f(x)$ and $t_n\xrightarrow{a.s}t$ as ${n \to \infty}$, is this always true that $f_n(t_n)\xrightarrow{a.s}f(t)$ as ${n \to \infty}$? Note that $t_n \xrightarrow{a.s} ...
0
votes
2answers
35 views

Characterizing Convergence of a Series

Let $\sum\limits_{n=1}^\infty$ $a_n$ be a convergent series of positive terms. What can be said about the convergence of the following series: $\sum\limits_{n=1}^\infty$ $\frac{a_1 + a_2 + ... + ...
0
votes
2answers
29 views

Convergence in probability, but not almost surely [closed]

Independent random variables $(Y_n)_{n≥1}$ are defined by $$P(Y_n = n) = 1/n$$ $$P(Y_n = 1) = 1 − 1/n$$ Show that, as $n → ∞, Y_n$ converges in probability but not almost surely. I am having ...
-1
votes
3answers
68 views

$f$ such that $\int_1^{\infty}f(x)dx$ converges, but not absolutely? [closed]

What's an easy example of a function $f$ such that $$\int_1^{\infty}f(x)dx$$ converges, but not absolutely?
2
votes
2answers
61 views

Proving the convergence/divergence of $\sum_{n=1}^\infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+…+\frac{1}{\sqrt{n}})$ [closed]

Do the following series converges? Why? $$ \sum_{n=1}^ \infty \frac{\cos\ n}{n} (1+\frac{1}{\sqrt{2}}+\cdots+\frac{1}{\sqrt{n}})$$
0
votes
4answers
29 views

Convergence when the comparison test cannot be applied

I had a standard problem in my textbook which was to determine the convergence of $\sum _{n=2}^\infty\frac{n^3+1}{n^4-1}$. To determine whether the series is convergent or not the standard solution ...
0
votes
3answers
28 views

How could we show that $s_n=2e^{(-n)}$, if $n$ is even & $-3\over n$ when $n$ is odd converges to $0$ as $n \to \infty$?

How could we show that $s_n=2e^{(-n)}$, if $n$ is even & $-3\over n$ when $n$ is odd converges to $0$ as $n \to \infty$ ? Using only the definition of convergence. So what I have tried is finding ...
0
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0answers
8 views

Trace of the heat operator $Z(t)=\sum_{m,n=1}^{\infty}e^{-\frac{\alpha_{m,n}^2}{r_0^2}t}$

I know that the spectrum of the disk of radius $r_0$ is $\lambda_{m,n}=\frac{\alpha_{m,n}^2}{r_0^2}$, where $\alpha_{m,n}$ is the n-th root of the Bessel's function of order $m$. I have to find the ...
0
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0answers
23 views

Pole Of Zeta function extension

Please advise on how to proceed with this? I don't know where to begin? Thanks. Show that the Riemann-Zeta function has a pole of order 1 at 1 after it has been extended to a holomorphic function on ...
0
votes
1answer
32 views

Bound for series

Please help. I am rather stuck with this problem (and our lecturer says she doesn't know how to do this either). It is part of a proof that the Gamma function is holomorphic, but was omitted from the ...
0
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0answers
25 views

Attempting to understand a theorem regarding subsequences and convergence

I am attempting to understand the following theorem. Theorem: Let ($s_n$) be a sequence. If $t$ is in $\mathbb{R}$ then there is a subsequence of ($s_n$) converging to $t$ if and only if the set ...
0
votes
1answer
17 views

Convergence of series

Can someone help me with this series? Let $C>1$ and $\alpha < 1$, does the series $\sum_{n = 0}^{\infty} C^{(\alpha^n)}$ diverges?
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votes
1answer
34 views

how to prove that $\sqrt(s_n) \to \frac{1}{2}$ as $n \to \infty$ given that $s_n \to \frac{1}{4}$? [closed]

how to prove that $\sqrt{s_n} \rightarrow \frac{1}{2}$ as $n \rightarrow \infty$ given that $s_n \rightarrow \frac{1}{4}$ ? only using the definition of convergence i.e. for all $\epsilon >0$ ...
1
vote
1answer
18 views

Annular regions in which the Laurent series converges

For the series $$\sum^\infty_{-\infty}\frac{z^n}{3^n + 1}$$ Determine the annular region in which this series converges. I understand that $\sum^\infty_{-\infty}\frac{z^n}{3^n + 1}$ can be split into ...
4
votes
3answers
44 views

Examples of sequences of positive terms $\{a_n\}$ such that $a_n^{1/n}\rightarrow l ~~\text{does not imply}~~ \frac{a_{n+1}}{a_n}\rightarrow l$

Give some examples of sequences of positive terms $\{a_n\}$ such that $$a_n^{1/n}\rightarrow l ~~\text{does not imply}~~ \frac{a_{n+1}}{a_n}\rightarrow l$$ If $a_n>0$ for all $n\in ...
3
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0answers
44 views

Compute $\sum_{n=1}^\infty e^{\pi^2(\frac{n^2}{a^2})t}$ [duplicate]

I have to compute $$Z(t)=\sum_{n=1}^\infty e^{-\lambda_nt}$$ with $\lambda_n=\pi^2\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)$. So $$\sum_{m,n=1}^\infty ...
0
votes
1answer
24 views

Squeezing theorem for proving convergence

I have the following series and want to prove it is converge using the squeezing theorem and root test $$\sum_{i=1}^\infty \frac{(-1)^n + 5}{3^n}$$ Just to bound it between two series and then use ...
1
vote
1answer
36 views

Prove that $\int _e^\infty \frac{\ln(x)}{x^p} dx$ is divergent for $p \le1$.

Prove that $\int _e^\infty \frac{\ln(x)}{x^p} dx$ is divergent for $p \le1$. So my textbook divides the problem into first case $p=1$ and integrates and cases $p<1$ in which it uses integration by ...
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vote
1answer
43 views

Convergence of $x_n=f\left(\frac{1}{n}\right), n\geq 1$

I have the following problem to solve: Let $f:]0,1[ \rightarrow R$ be a differentiable function over $]0,1[$, with $|f'(x)|\leq 1, \forall x \in ]0,1[$. (a) Show that the sequence ...
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votes
2answers
21 views

Decide if the following series is converge [closed]

Im trying to prove if the following series is converge - I tried the ratio test but couldnt calculate the limit - $$\sum_{i=1}^\infty \frac{(2n)^{n+2}}{(n+1)!}$$ Thanks for helping!
0
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2answers
66 views

Bessel's function in spectral geometry [closed]

I have to compute $Z(t)=\sum_{n=1}^\infty e^{-\lambda_nt}$, $t \in \mathbb{R}_{>0}$, with $\lambda_n=\pi^2(\frac{m^2}{a^2}+\frac{n^2}{b^2})$. So $\sum_{m,n=1}^\infty ...
5
votes
1answer
40 views

Norms inequality in a sequence space

Let $1 \leq p<q \leq \infty$ (p an q are not related) Let $\Phi$ be the vector space of all sequences with at most finitely many nonzero elements, meaning $\Phi=\{\{x_n\}_{n=1}^\infty|$ there is ...
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vote
1answer
22 views

Is there a rearrangement theorem for conditionally convergent improper integrals?

The famous Riemann rearrangement theorem states that for a conditionally convergent real number series, we can rearrange the order of summation to make it converge to any prescribed number in the ...
0
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1answer
34 views

Uniformly bounded sequence of analytic functions in the unit disk

Suppose $\{f_n\}$ is sequence of analytic functions that is uniformly bounded in the open unit disk and for every positive integer $k$, $f_n(\frac{1}{1+k})\to 0$ pointwise. Then, $\{f_n\}$ converges ...
0
votes
1answer
13 views

Convergence in mean square and almost surely

Given the sample space [0,1] and the uniform probability measure P(.), random variables $(X_n)_{n\geq1}$ are defined by How do I $X_n$ converges almost surely as n tends to infinity and also in ...
1
vote
2answers
37 views

Convergence of the sequence $2n\times\sin(\frac{1}{n-1})$?

I have the sequence $$\{2n*\sin(\frac{1}{n-1})\}$$ and I'm supposed to see if this sequence converges. By transforming this sequence into a function of $f(x)$ and applying L'Hopital's rule to the ...
0
votes
0answers
18 views

How to formally justify matrix manipulation in countable-state Markov chain

I have a Markov chain with transition probabilities $t_{i,i+1} = \binom{k+i}{k}^{-1}$ and $t_{i,0} = 1-t_{i,i+1}$, i.e. we have an absorbing chain with absorption probability approaching one as $i ...
0
votes
0answers
15 views

Understanding the Strong Law of Large Numbers

Strong law of large numbers (SLLN) says if $X_1, X_2, \dots$ are iid random variables with expectation $\mu$, then $\bar{X}_n \to \mu$ almost surely, or $$P(\lim_{n\to \infty} \bar{X}_n = \mu)=1.$$ ...
0
votes
1answer
16 views

Convergence of $\int_0^1 x^p ln^q \left(\frac{\ 1}{x}\right)$

So far, I determined that the integral converges for every $q>p+1$. I noticed that for example for the values $p=5, q=3$ the integral still converges. There are some values for which the integral ...
1
vote
2answers
36 views

Does $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^n$ and $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^{n^2}$ converge?

My task is this: Determin whether $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^n$ and $\sum_{n=1}^\infty \left(\frac{1}{n} - 1\right)^{n^2}$ converge or diverge. My thoughts: For large $n$ one ...