Convergence of sequences and different modes of convergence.

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1answer
22 views

Find convergence domain of integral

I need to find convergence domain of $$\int_1^2 \! \frac{\ln(x-1)}{(4-x^2)^p} \, \mathrm{d}x$$ I've tried to use estimates like $\frac{\ln(x-1)}{(4-x^2)^p} < \frac{1}{(4-x^2)^p}$ and change of ...
1
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0answers
24 views

Show that $C^1([a,b])$ is a complete space w.r.t. some special metric [duplicate]

Let $C^1([a,b])$ denote the space of all continously differentiable functions on $[a,b], a,b\in\mathbb{R}$. On this space, define the following metric: $$ d(f,g)=d_{\infty}(f,g)+d_{\infty}(f',g'), $$ ...
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3answers
36 views

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$

Prove that functional series $\sum_{n=1}|sinx|^{\sqrt{n}}$ is convergent when $x \in (-1,1)$. I do not know if that's so easy that I'm simply missing something, but I can't find any criterion which ...
1
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2answers
72 views

Solving equations like $xe^x = c$ via functional iteration

Yesterday I randomly thought of solving $xe^x = c$ via functional iteration (FI) after manipulating the equation into a form "$x = \cdots$" that gives the 'fastest' convergence rate regardless of the ...
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0answers
27 views

find $\text{limsup} \dfrac{X_n}{\ln{n}}$? how can i apply Borel-Cantelli here?

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{-|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
2
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3answers
47 views

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
2
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1answer
34 views

Integrating variation of error function: $\int_1^2e^{-nx^2} dx$

Show that $$\lim_{n\to\infty} \int_1^2e^{-nx^2} dx = 0.$$ After much googling, I learned that I am working with a variation of the error function! Yay. I've never heard of it in my life and I ...
0
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1answer
37 views

Shouldn't all alternating series diverge by the diverge test?

An Alternating Series, as defined in my textbook, is of the form $\sum (-1)^n b_n$. If we look at the nth term, the series doesn't appear to converge. If n is odd, the nth term is negative; if it's ...
2
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0answers
18 views

Prove $\sum_{k=1}^\infty k^{-p}f(kx)$ converges absolutely almost everywhere, where $p>0, f \in \mathcal{L}^1(\mathbb{R})$.

What I've done: $$ \int_\mathbb{R} \sum_{k=1}^\infty k^{-p}|f(kx)| = \sum_{k=1}^\infty \int_\mathbb{R} k^{-p}|f(kx)|dx = \sum_{k=1}^\infty k^{-p}\int_\mathbb{R} k^{-1}|f(y)|dy = ...
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24 views

series convergence help using tests [closed]

Use any theorems or properties of series. This was a question on my homework and I received 0 points. I need help with the entire question. I originally tried to compare part (a) to the ...
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0answers
23 views

show that $U_n$ converges to $0$ in $L^1$ and almost surely.

let $(X_n)_{n\geq1}$ be a sequence of independent random variables. Suppose that the density function of $X_n$ is: $$ f(x)=\dfrac{1}{2}.e^{|x|} \quad x \in \mathbb{R} \quad \forall n \quad ...
2
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0answers
20 views

Almost sure convergence and limes superior

I'm trying to prove the following exercises and I don't know if my attempts are correct. A sequence of real random variables $(X_n)$ almost surely converges to $X$ if and only if for every $\epsilon ...
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1answer
25 views

Relation between $\lim_{n \to \infty}\int_{I}f_n(x)\:dx$ and $\int_{I}\lim_{n \to \infty}f(x)\:dx$ [closed]

Relation between convergence and integration of sequence of a function. Let $f_n$ be a sequence of integrable functions defined on an closed interval with $$f_n(x) \to 0$$ on this interval ...
2
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2answers
31 views

Showing a recursively defined sequence is convergent

For $a_1=1$ and $a_{n+1} = 1 + \frac{a_n}{3+n}$, I want to show that the sequence $a_n$ converges. I will use the Monotone Convergence Theorem. Of course, the sequence is bounded below by $1$. Now I ...
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0answers
27 views

Is there a way to calculate RMS value continuously?

Using that the RMS by definition is: $\sqrt {\int_0^T\frac 1T*f(t)^2dt} $ which can be calculated by using Riemann sums in the following way: $\sqrt {\frac 1N\sum_0^Nf[i]^2} $ I've tried that in ...
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4answers
55 views

Why does the sequence $a_n = (n^2)(1 - \cos(\frac{4.4}{n}))$ converge to 9.68?

Find the limit of the sequence whose terms are given by $a_n = (n^2)(1 - \cos(\frac{4.4}{n}))$. The given answer for this problem is $9.68$. What rules about sequences, and steps, should be taken ...
1
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1answer
56 views

$a_n = b_n -b_{n-1}$ Prove that $\sum_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists

Let $\{b_n\}$ be a sequence Let $a_n = b_n - b_{n-1}$. Prove that $\sum\limits_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists. I am extremely stuck on this homework problem and ...
2
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1answer
63 views

Prove that $\frac{n^n}{(n + 1)^n}$ converges to $\frac{1}{e}$

Using the formal definition of sequence limits, I would like to prove that for: $$ a_n := \frac{n^n}{(n + 1)^n} $$ it is: $$ \lim_{n\to\infty} a_n = \frac{1}{e}. $$ Thus, it remains to show ...
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0answers
70 views

Is $n^{1/10}$ a cauchy sequence?

So, I stuck at a step to prove that $n^{1/10}$ is not a cauchy sequence. So, $$|y_n - y_m| = |n^{1/10} - m^{1/10}|$$ So, now how is the next step to show, that it is not a cauchy sequence?
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2answers
44 views

Theorem 3.19 in Baby Rudin: The upper and lower limits of a majorised sequence cannot exceed those of the majorising one

Here is Theorem 3.19 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Let $\{s_n \}$ and $\{t_n \}$ be sequences of real numbers. If $s_n \leq t_n$ for $n \geq N$, ...
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0answers
20 views

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 ...
0
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2answers
47 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 ...
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2answers
46 views

Finding an integral involving logarithmic functions: $\int_0^\infty\frac{1}{z[\ln(z)]^2}dz$ [closed]

Finding the following integral: $$\int_0^\infty \frac{1}{z[\ln(z)]^2} dz$$
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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 ...
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0answers
14 views

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
49 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 ...
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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 ...
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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= ...
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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 ...
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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 ...
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0answers
19 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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18 views

For each of the following space, answer these questions [closed]

: a) Which sequences converge to which points? b) Is X first countable? c) Does the result of Theorem 10.4 hold( Theorem 10.4 . If X is a first-countable .space and Ε in X, then x belong E iff ...
1
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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 ...
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0answers
22 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
40 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 \ ...
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0answers
13 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 ...
2
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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
vote
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 ...
0
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0answers
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 ...
1
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3answers
49 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
votes
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 ...