Convergence of sequences and different modes of convergence.

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1answer
19 views

Let {xn} be a sequence that has two subsequences converging to different limits. Prove {xn} is not convergent. [on hold]

I can't use that if {xn} is a convergent sequence, then any subsequence {xni} is also convergent and their limits are equal.
0
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0answers
13 views

Finding where a complex series converges absolutely, uniformly.

I need to figure out where the series converges absolutely and uniformly. I know that once I have absolute convergence on a region, then I know I also have uniform convergence on that region, so I ...
1
vote
1answer
30 views

Using MCT twice to show the limit of an integral depending on $x$ and $n$

So I have $\displaystyle\lim_{n \to \infty} \int^{n^2}_0 e^{-x^2} n \sin\left(\frac{x}{n}\right) dx$. I'd like to apply the MCT but the trouble is there is a limit which also depends on $n$ So I ...
4
votes
1answer
49 views

Check series for convergence

$$ \sum_{n = 1}^\infty \sin(n)\sin\left(\frac{(-1)^n}{n^{1/4}}\right) $$ I have no idea how to deal with it.
0
votes
2answers
59 views

Proving that the series $\sum [\log(2n+1)-\log(2n)]$ diverges.

Let $f(n)=\log(2n+1)-\log(2n)$. Using the Cauchy's condensation test we have: $$2^nf(2^n)=2^n[\log(2\cdot2^n+1)-\log(2\cdot2^n)] = ...
3
votes
2answers
39 views

Check for convergence

$$\sum_{n = 2}^\infty (-1)^n \sin\left( \frac{\sin(3n)}{\sqrt{n}} + \frac{1}{n (\ln n)^2}\right)$$ I tried to use Maclaurin series, but failed to evaluate little-o.
0
votes
2answers
25 views

Convergent sequences must have bounded range?

I am currently reading Baby Rudin and I am having trouble understanding why convergent sequences must have a bounded range. Specifically, I am thinking of the following counterexample: $f(n)=1/(n-1)$ ...
0
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0answers
38 views

Series Convergence by Comparison [on hold]

I've been working on some calculus problems and am struggling to understand what I can compare this series to in order to prove convergence: $$ \sum_{n=0}^{\infty} \sin^2(\pi n) $$ I know that all ...
1
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0answers
14 views

Fix a typo involving the Lobachevsky function in Thurston's notes

I believe that there is a typo in these great notes Thurston's Three-Dimensional Geometry and Topology, Volume 1 (Princeton University Press, 1997), Chapter 7 that is provide us by MSRI, in the ...
7
votes
1answer
131 views

Prove that sum is convergent

How to prove that the following sum is convergent? $$\sum_1^\infty\frac{\sin(n + \ln{n})}{n}$$ I tried to use formula $$\sin(n+ \ln{n}) = \sin{n}\cos \ln{n} + \sin \ln{n}\cos{n}$$ and $$\sum_1^N ...
1
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0answers
31 views

About сonvergence of partial sums of basis of Banach space

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i ...
3
votes
0answers
15 views

weak convergence and unbounded functions with bounded moment

I want to prove the following: Given a topological space (it is a Lusin space, but I think that does not matter) $\Omega$, a function $f \in C(\Omega,\mathbb{R})$ and a sequence of Radon measures ...
1
vote
1answer
65 views

Proving that $\lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$

Prove that $\displaystyle \lim_{n \to \infty} (1+ \frac{i}{n})^n = e^i$ I was trying to proof in the same way of $\lim (1 + \frac{1}{n})^n = e$, but I couldn't proceed this way. Can someone give me a ...
2
votes
2answers
36 views

Taking two times the sequential closure of the set of continuous functions in the topology of pointwise convergence?

Consider the unit interval $I=[0, 1]$ and assume that the function $f\colon I\to \mathbb R$ satisfies $$ f(t)=\lim_{n\to \infty} f_n(t), \qquad \text{for all }t\in I $$ where $$ f_n(t)=\lim_{j\to ...
1
vote
3answers
44 views

Finding a formula for a kth element in a sequence

I've setup a recurrence relation as part of a numerical analysis problem, and found that $$x_{n+1} = \frac{x_n+1}{2}$$ The notes then say that for $x_0=0$, it is easy to show that $$x_k = 1 - ...
3
votes
0answers
75 views
+50

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 are given a distribution $p_{z}(k)$ over the whole $\mathbb{Z}^+$. We are interested in approximating $p_v(v)$ over ...
0
votes
1answer
15 views

Uniform convergence of supremum

If a sequence $\{f_n\}$ converges uniformly to a limit $f$ on the domain $D$, then the sequence $\{M_n\}$, with $M_n = \sup_{x} |f_n(x)-f(x)| $, converges to zero. So what I thought was since ...
0
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1answer
32 views

Convergence of partial sums of basis vectors in banach space

Let $B$ is some separable Banach space with Schauder basis $\{e_i\}_{i=1}^\infty \subset B$. Let $\{\alpha_i\}_{i=1}^\infty$ - is some sequence of complex numbers. Let $p_n = \sum_{i=1}^n \alpha_i ...
-1
votes
1answer
29 views

Integral convergence $\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}}$

I've tried to partial this integral from 0 to 1, 1 to e, and e to infinity. $$\int_{2}^{\infty} \frac{\cos x}{\sqrt[3]{\ln x}} dx$$
2
votes
2answers
861 views

Use the ϵ-N denition of limit to prove that lim[(2n+1)/(5n-2)] = 2/5 as n foes to infinity

Use the ϵ-N definition of limit to prove that lim[(2n+1)/(5n-2)] = 2/5 as n goes to infinity. The way I do it is Let ∊ > 0 be given. Notice N ∈ natural number (N) which satisfies {fill this box ...
3
votes
2answers
41 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 ...
2
votes
3answers
88 views

Series convergence $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$

Series $A = \sum_{n=1}^\infty\frac{1}{\ln(n+1/n)}$ diverges by the comparison test (wolfram). I want to compare $\sum_{n=1}^\infty\frac{\sin^4n}{\ln(n+1/n)}$ with series $A$. How can I prove that ...
-2
votes
1answer
22 views

Convergence pw if converges in Lp space

Let $p\in[1,\infty]$ be given. If $f$ and $g$ are non-negative analytic functions such that the following holds: \begin{equation} ...
1
vote
3answers
72 views

Limit of $\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}$ when $n\to\infty$

I have to show the convergence of the series $$\lim\limits_{n \to \infty}a_n=\sum \limits_{k=1}^{n} \frac{2^kn+2n^2+k}{2^{k+1}n^2+2^kk}.$$ I am quite sure that the limit is 1.5. I wanted to show this ...
0
votes
1answer
48 views

Are these integrals convergent?

Recently I've come across two integrals that seemed hard to check for me. Here they are: $$\int_0^\infty \frac{x \sin \ln x}{x^2 + \cos x} \, \mathrm{d}x$$ And another: $$\int_1^\infty \frac{\sin \ln ...
2
votes
1answer
22 views

Convergence of a Fourier series to a point

Consider the function $f\left(x\right)=1+x$, $x \in \left[-\pi,\pi\right]$ I have calculated its Fourier series to be $$f\left(x\right)=1+2\sum^{\infty}_{n=0}\dfrac{\left(-1\right)^{n+1}}{n}\sin ...
0
votes
3answers
29 views

Proving a subsequence doesn't converge

When I want to prove that a sequence doesn't converge by showing that it's subsequence doesn't converge , can i use the limit comparison test? (Usually used for series) . for example - $$ \sum_{n ...
1
vote
1answer
65 views

Convergence of $g(x)\cdot f(x)$

Let $g(x)=\frac{1-e^{-x^2}}{x^2}$ for $x \neq 0$,$g(0)=1$ and $f(x)=e^{-(x-n)^2}$. You can assume that g(x) is continuous and bounded with maximum 1 in x=$0$. Show that $\sum_{n=1}^{\infty}g(x)\cdot ...
3
votes
1answer
19 views

Theorem 3.44 in Baby Rudin: Can we replace the coefficients with their absolute values?

Here's Theorem 3.44 in the book Principles of Mathematical Analysis by Walter Rudin, third edition: Suppose the radius of convergence of $\sum c_n z^n $ is $1$, and suppose $c_0 \geq c_1 \geq c_2 ...
2
votes
3answers
70 views

How to show $\lim _{x\to 0}\:\frac{\sin\left(\frac{1}{x^2}\right)}{x^2}$ does not exist?

$$\lim _{x\to 0}\:\frac{\sin\left(\frac{1}{x^2}\right)}{x^2}$$ I my intuition is telling me this limit does not exist as $\sin$ will be oscillating but will stay bounded and then will blow up as ...
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0answers
13 views

Examples 3.35 (a) and (b) in Baby Rudin: Limit Superior and limit inferior of a couple of sequences

This question is related to Examples 3.35 (a) and (b) in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition, p. 67. Let us consider the series $$ \frac 1 2 + \frac 1 3 + ...
-1
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1answer
17 views

Bounded convergence theorem - 2M

Can someone please help me with where the 2M is coming from?
4
votes
3answers
78 views

What is $\lim_{x\to \infty} 2\sqrt{x}- \sum_{n=1}^x {1\over \sqrt{n}}$? [duplicate]

I ask this because I noticed the partial sum $\sum_{n=1}^x {1\over \sqrt{n}}$ is very close to $2\sqrt{x}$, so close in fact that it appears their difference approaches a constant value, like $H_x$ ...
0
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0answers
29 views

Power Quantum Series And It's Sum.

Let $I$ be the interval $(-\theta, \theta), \theta=q^{\frac{1}{1- n}}$, $n\in 2\mathbb{N}+1$ and $q\in (0,1)$ are fixed. Define a function $h(t):=qt^{n}$. One can see that the $k$-th order iteration ...
0
votes
1answer
36 views

Convergence of $\sum_{k \geq 1} e^{-tk} \cos kz$

I would like to find the convergence of the series $\sum_{k \geq 1} e^{-tk} \cos kz$. Clearly, this series converge in using the comparison test or the integral. How could I get an explicit function ...
1
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0answers
49 views

Show that $\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ [duplicate]

$\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$ Tried: here suppose N is poission distribution with parameter n $\lim_{n\rightarrow \infty} \sum_{i=0}^{n} ...
1
vote
2answers
33 views

What's wrong with my radius of convergence test?

Given $\sum \limits _{n=2} ^\infty \frac{(\ln x)^n} n$, find its radius of convergence $R$. Using the ratio test, I arrived at $$|\ln x| < 1 \implies e^{|\ln x|} < e^x \implies |x| < e ...
0
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1answer
14 views

Determine whether the fourier series converges

I have calculated the Fourier Series of $g\left(x\right)=x$ on $\left(-\pi,\pi\right]$ extended periodically to $\mathbb{R}$ to be ...
0
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2answers
32 views

How can i chech the convergence of $ \sum_{i=0}^{ \infty} \frac{ \theta^n}{n^r} $ , $r >1$ and $ \theta \in (0,1) $?

How can i chech if the serie of $ \sum_{n=1}^{ \infty} \frac{ \theta^n}{n^r} $ , $r >1$ and $ \theta \in (0,1) $ is converge?
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2answers
40 views

For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N

In order to prove: For $f:M\to N$ to be continuous its sufficient that $x_n\to a\implies f(x_n)_n$ is convergent in N I'm supposing that $x_n$ is convergent, that is: $$\forall \epsilon>0, ...
0
votes
1answer
27 views

Uniform convergence of a function composition

Let $n \in \mathbb{N}, f_n(x)=e^{-(x-n)^2}$ and $g(x)=\Bigg\{\begin{array}{ll} 1 & x=0 \\ \frac{1-e^{-x^2}}{x^2} & x \neq 0 \\ \end{array} $ You can assume that g is continuous ...
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votes
2answers
31 views

What is the convergence value of series $\sum_{i=1}^{\infty} i^2 * (0.4)^i$

One technique to cope with some series is using derivation of a geometry series. But in this case I think $i^2$ makes this technique useless. Any idea would be appreciated.
0
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1answer
20 views

Is the integral test for convergence still applicable?

$$\sum _{n=0}^{\infty \:}\left(n\ e^{-n^2}\right)$$ Can I still use the integral test to determine whether this series converges or diverges given that $f(x) = x\ e^{-x^2}$ is not decreasing on the ...
0
votes
1answer
42 views

Closed form of a series

I am looking for a closed form of the following convergent series: $$\sum_{n=0}^\infty \frac{(-\lambda^2)^n}{(6n+i)!}$$ For the case of $i=0$, the answer is ready, but when $i=1,2,3,4,5$, everything ...
3
votes
1answer
26 views

Absolute convergence of ordinary Dirichlet series

I am currently reading Serre's 'A course in Arithmetic' and I have a question about proposition 8 of section 2.4 (but I think the question can be answered without knowing the book.) The proposition ...
2
votes
3answers
77 views

Check convergence of $\sum^{\infty}_{n=1} \frac{2^{n} +n ^{2}}{3^{n} +n ^{3}}$

Zoomed version: $$\sum^{\infty}_{n=1} \frac{2^{n} +n ^{2}}{3^{n} +n ^{3}}$$ So, I've seen similar example at Convergence or divergence of $\sum \frac{3^n + n^2}{2^n + n^3}$ And I liked that answer : ...
0
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0answers
23 views

A complex sequence of functions $(f_n)$ is continuously convergent iff it's compactly convergent against a continuous function

Let $G \subseteq \mathbb{C}$ be a region in $\mathbb{C}$, i.e. $G$ is open, nonempty and connected, and let $f_n: G \to \mathbb{C}$ be a sequence of complex-valued functions, with $n \in \mathbb{N}$. ...
1
vote
1answer
60 views

What sum to $\sum_{n=0}^\infty\frac{x^n}{(n+1)!}$ in its convergence radus?

My task is this: Find the sum to $$\sum_{n=0}^\infty\frac{x^n}{(n+1)!}.$$ in its convergence radus. My work so far: By ration test we get ...
0
votes
1answer
22 views

Convergence of $\sum_{k=1}^\infty \frac{1}{k^{µ(k)}}$

For which $\alpha$ and $\beta$ is the sum $$\sum_{k=1}^\infty \frac{1}{k^{µ(k)}}$$ $$µ\left(k\right) = \left\{ \begin{array}{lr} \alpha & : k\ is\ even\\ \beta & : k\ is\ ...
1
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0answers
23 views

Monotone Convergence Theorem in Measure Theory.

My textbook defined M.C.T. by for $\{f_k\}$ be a sequence of measurable functions on $E\subset\mathbb{R}^n$, If $f_k\nearrow{}f~~a.e.$ on $E$ and there exists $\phi\in{}L(E)$ such that ...