Convergence of sequences and different modes of convergence.

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3
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0answers
48 views

Does the series $\sum_{k=1}^{\infty} \left[\ln\left(1+\frac{x}{k}\right) - \frac{x}{k} \right]$ converge?

I tried all the theorems, that I knew in analysis, to know if the mentioned series converge but none of them is relevant except one: The Ratio Test for Series, but unfortunately this is not working as ...
0
votes
1answer
18 views

Is a convergent power series on an open set continuous on that set?

Question in the title. If a power series $f(x)$ is pointwise (or if this is too weak, uniformly) convergent for every $x$ in an open set $U$ in the reals, is it a continuous function of $x$?
7
votes
5answers
406 views

Giving a specific example of a positive sequence increasing to 1 and with its partial products having a positive limit

In my real analysis class I was asked this which got me stick: Is there an example of a sequence of real positive numbers increasing to the limit 1 $ \{ a_n \}_{n=1}^{\infty} $ such that the ...
2
votes
2answers
48 views

(Correct Proof?) Show that every convergent sequence $(\mathbf{x}_{k})$ in $\mathbb{R}^{n}$ is bounded.

Now, this is a pretty simple proof and I just wanted some more experienced members here to have a look at it and maybe give me feedback on my proof idea for the statement in the title. I also found ...
0
votes
4answers
43 views

Factorial series with ratio test inconclusive

I have to determine the convergence of this series: $\displaystyle\sum_{n=1}^\infty {2n! \over (n+3)! }$ The ratio test gives 1, so I don't know how to solve it. Wolfram alpha suggests me to use the ...
-1
votes
2answers
58 views

Determine whether or not the following series is convergent $\sum_{n=1}^\infty \frac{1}{n^n}$ [closed]

Determine whether or not the following series is convergent $$\sum_{n=1}^\infty \frac{1}{n^n}$$ What test should I use to approach this question?
0
votes
1answer
25 views

Laurent series of $f(z) = \frac{1}{z^2-z}$ centered at $z= -1$ and converges at $z=-1/2$

I need to compute the Laurent series expansion of the following function - $$f(z) = \frac{1}{z^2-z}$$ centered at $z= -1$ and converges at $z=-1/2$ . I tried this problem using substitution $w=z+1$. ...
0
votes
2answers
42 views

Is a subspace of a topological space always closed?

Let $(X,T)$ be a topological space where $X$ has a vector space structure and let $V \subset X$. Is it true that if $V$ is a vector subspace of $X$ then $V$ is closed in the topology $T$? Sorry if ...
-1
votes
1answer
47 views

Determine whether or not the following series are convergent $\sum_{n=1}^\infty n\sin(\frac{1}{n})$

Determine whether or not the following series are convergent$$\sum_{n=1}^\infty n \sin\left(\frac{1}{n}\right)$$ How do I go about using the nth term test to prove this without the use of ...
0
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2answers
52 views

Normal convergence implies uniform absolute convergence but not the other way round

How do I show that normal convergence of a series implies uniform and absolute convergence? So, a series $f_1+f_2+...$ of functions $f_n:D\rightarrow\mathbb{C}, D\subset\mathbb{C}$ is normally ...
-1
votes
1answer
48 views

Prove $a_n = \sin(n\pi /2)$ does not converge to $0$

Not sure how to prove it doesn't converge to $0$. Prove by contradiction? How do I assume that the sequence converges to $0$ since if I do the partial sums it does end up adding to $0$.
0
votes
0answers
19 views

Prove by the definition of convergence that the sequence an converges to 1.

I know the definition of convergence as a sequence an approaches the limit if for every $\epsilon > 0$, there exists $N \in\mathbb N$ such that $|a_n - L| \lt \epsilon$. How do I prove with this ...
1
vote
2answers
28 views

Lower bound for $2+ \cos(t)$

I was doing this question on convergence of improper integrals where in our book they have used the fact that $2+ \cos(t) \ge1$. Can somebody prove this?
0
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0answers
13 views

Convergence of Sums With Logarithmic Numerator

Let $S \subseteq \mathbb{N}$ have the property: $\displaystyle \sum_{n=1}^\infty \frac{\ln (s_n)}{s_n}$ converges. I'm just wondering if there are any known theorems out there which allow us to derive ...
0
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1answer
37 views

If S = {1/n | n ∈ ℕ}, what is inf(S)? [duplicate]

I believe the answer is 0 but I'm not really sure how to prove it...does it involve using epsilon?
0
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1answer
19 views

${f_n}$ differentiable and $f'_n$ converges uniformly on $[a,b]$. How to show $f_n(x) - f_n(a)$ also converges?

I tried using 2 mean value theorems but I got 2 different x values: $(x-a)f_n'(c)$ and $(x-a)f_m'(d)$ so I couldn't make use of $f'_n$ unif convergence. What should I tweak?
-1
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0answers
9 views

multiple convergences

Suppose $X_n \xrightarrow{a.s.} Y_n$ and $Y_n \xrightarrow{a.s.} c$, where $c$ is a constant. Then can we say that $X_n \xrightarrow{a.s.} c$? What about if we replace a.s. with convergence in ...
1
vote
1answer
45 views

$X_n$ converges to zero in probability

Suppose $X_n = 1$ with probability $\frac{1}{n}$ and zero elsewhere. I am trying to find out whether $X_n$ converges to $0$ in probability. I understand I need to prove $\lim\limits_{n \to \infty} ...
1
vote
2answers
32 views

Check: Radius of Convergence of the Sum of these Complex Taylor Series

I just found the following Taylor series expansions around $z=0$ for the following functions: $\displaystyle \frac{1}{z^{2}-5z+6} = \frac{1}{(z-2)(z-3)} = \frac{-1}{(z-2)} + \frac{1}{(z-3)} = ...
0
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0answers
33 views

How to check if this series is convergent? [duplicate]

Check if this series is convergent: $\sum_{}^{} \left( \sqrt[n]{a}- \frac{\sqrt[n]{b}+\sqrt[n]{c}}{2} \right)$, a,b,c>0
-1
votes
0answers
14 views

gauss seidel covergence

I'm very poor at math, I have such Matrix to check if gauss-seidel method is covergence or not. I found lots of examples to check if is it covergence or not, but all of them have answer b, I did not ...
0
votes
2answers
56 views

If $a_n \leq cb_n$, Then if $\sum b_n$ converges $\sum a_n$ converges.

Suppose that $\sum a_n$ and $\sum b_n$ are series of positive terms and that there is a positive number $c$ s.t. $a_n \leq cb_n \forall n$. Show that if $\sum b_n$ converges, then $\sum a_n $ ...
2
votes
3answers
97 views

Convergence or divergence of $\sum \frac{3^n + n^2}{2^n + n^3}$

I need to conclude if the following series is convergent $$ \sum_{i=1}^\infty \frac{3^n+n^2}{2^n+n^3}. $$ Can I get a hint? I tried to calculate $\dfrac{a_{n+1}}{a_{n}}$ and to see if the series is ...
0
votes
1answer
29 views

Convergence of a complex series for some values of z

Find all $z \in \mathbb{C}$ for which the series $$\sum_{n=0}^\infty(-i)^nz^n$$ converges. I'm not really sure where to start. I know that most convergence tests for real numbers also hold for ...
1
vote
1answer
27 views

Convergence of an improper integral with a parameter

I am to test the convergence of the improper integral $$ \int_{0+}^{1-} \frac{\ln(x)}{(1-x)^a} dx$$ with the parameter $a \in R$. I have some trouble doing this so I'd appreciate a full explanation so ...
1
vote
1answer
20 views

Rate of convergence of a solution to an equation with a free parameter

Suppose that $\epsilon>0$ is a solution to the following equation: $$\epsilon^2-a(m)\ln\epsilon-b(m)=0,$$ where $a(m)\to 0^+$ and $b(m)\to 0$ as $m\to\infty$. Suppose that a solution ...
5
votes
0answers
136 views

When does $x^{x^{x^{…}}}$ converge? [duplicate]

Provided that $x$ is positive, when does the following converge? $$x^{x^{x^{x^{...}}}}$$ So here is my work, but I don't really know if this is correct. Could anyone shed some light on this? I ...
27
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5answers
1k views

What happens if I repeatedly alternately normalize the rows and columns of a matrix?

Here is an algorithm: input matrix M (in-place) divide each row of M by its norm divide each column of M by its norm repeat What will M look like after this has been repeated many times? Can we ...
1
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3answers
51 views

Does $a_k$ diverges

I'm not sure, If I'm doing the convergence test correctly. From given, $\sum_{k=1}^{\infty} b_{k}$ diverges and $\sum_{k=1}^{\infty} \frac{a_{k}}{b_{k}}=2$ I have to check whether ...
0
votes
0answers
18 views

implication of continuity of function in proving convergence of sequence

I am reading a paper named "Asynchronous Broadcast-Based Convex Optimization Over a Network" by Nedic and I am confused about a part I attach as follows. I don't know the rational deducting from the ...
0
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1answer
16 views

Convergence of functionals in the dual space

Let $M\subseteq X$ be a subset of a normed space. I have been asked to show that the annihilator of $M$,$M^a$ is closed. To do this I assume that it isn't closed. I.e there exists some functional ...
0
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1answer
9 views

Need some help in understanding a root test with given answer

Trying to proof the convergent of the series in the box by root test, the answer is as shown. However two parts which I do not understand why does the equation became $(1+1/n)^n$ instead ...
2
votes
1answer
30 views

Is this proof of divergence of an alternating series correct?

Determine whether the series $\sum_{r=1}^\infty (-1)^{r-1}(\sqrt{r+1}-\sqrt{r})$ is convergent, absolutely convergent, or divergent. The way the textbook did it is that they let $b_r = ...
1
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1answer
15 views

transformations divergent series

I need help, If $\forall n\in \mathbb{N}, x_n> 0 $ and $ \sum_{n=1}^{\infty} x_n$ is divergent then $ \sum_{n=1}^{\infty} \frac{x_n}{1+x_n}$ divergent? thanks for the help
0
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0answers
20 views

Almost sure limit of a sequence

I have a random variable $X$, which is uniformly distributed on $[0,1]$. The underlying probability space is as usual $(\Omega,\mathcal{F},P)$. For every $t$ with $0\leq t < 1$, I define a random ...
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0answers
14 views

arithmetic mean of a bounded sequence converges? [duplicate]

I have learnt that the means of a convergent sequence also converges. However, I am wondering whether can we say the means of a bounded sequence converge? or any other counterexample? Let ...
3
votes
2answers
20 views

Convergence of series $\sum_{n = 1}^\infty \exp(-n^a)$, when $0 < a < 1$.

Consider the non-negative series $$\sum_{n = 1}^\infty e^{-n^a}, 0 < a < 1.$$ If $a = 0$, the series is divergent, and if $a \geq 1$, by root test, it is convergent. Root test doesn't give ...
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3answers
42 views

$1/z_n$ converges to 0 if and only if $z_n$ diverges. For complex numbers.

In real analysis there was an easy property that converted limits to infinity in limits at zero. More precisely, $1/z_n$ converges to 0 if and only if $z_n$ diverges (this is converges to infinity). ...
0
votes
1answer
26 views

Sequence of functions that fails certain conditions of Arzela-Ascoli theorem

For a closed, bounded interval $[a,b]$, let $\{ f_{n}\}$ be a sequence in $C[a,b]$. If $\{f_{n}\}$ is equicontinuous, does $\{f_{n}\}$ necessarily have a uniformly convergent subsequence? I would ...
3
votes
2answers
52 views

What does the series $\sum_{n=2}^\infty \frac{2}{n^3-n}$ converge to?

I know that the series converges. My questions is to what. I tried seeing if it was a telescoping series: $\sum_{n=2}^\infty \frac{2}{n^3-n} = 2\sum_{n=2}^\infty (\frac{1}{n^2-1}-\frac{1}{n})$ but it ...
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3answers
61 views

How do I show the series $\sum{\frac{1}{\log(n)^{\log(n)}}}$ converges? [closed]

How do I show the series $\sum{\dfrac{1}{\log(n)^{\log(n)}}}$ converges?
0
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0answers
41 views

Uniqueness of solution to integral equation with “endogenous” kernel

Let $x,y\in C$ and consider the functional equation $T:y\mapsto x$ implicitly defined by the following integral equation: $$ x(t) = \int_{-\infty}^\infty K(x(t),t,s) y(s) \, ds, $$ with $K$ given. ...
0
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1answer
25 views

Radius of Convergence for a complex sin function

I encountered the following power series, and while I know a couple of ways to determine radius of convergence, I wasn't able to figure out how to evaluate the appropriate limit to get said radius. ...
0
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1answer
29 views

From known power series deduce the power series expansion of $ln(5-x)$ and infer the general term and radius of convergence

From known power series deduce the power series expansion of $ln(5-x)$ and infer the general term and radius of convergence. Now I said: ...
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1answer
34 views

Under what conditions will $f(x)^n$ converge pointwise and uniformly?

Let $f(x)$ be a continuous function on $[0,1]$. Under what conditions on $f$ will the sequence $f_n(x)$ = $(f(x))^n$ converge uniformly and pointwise?
0
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1answer
18 views

convergence, bounded sequences, and limits

Just want to get this straight in my head, so when doing proofs, proving that a sequence is convergent and proving if it is bounded is pretty much doing the same thing. They both use the same ...
2
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1answer
40 views

Determining convergence of $\frac{n}{e^n}$

I am trying to determine if $\sum_{i=1}^\infty \frac{n}{e^n}$ is converging This is what I have so far $\sum_{i=1}^\infty \frac{n}{e^n}$ converges by the geometric series test since $\frac{1}{e}$ ...
0
votes
2answers
64 views

Prove that $\lim_{n \rightarrow \infty}\sqrt[n]{n}=1$ [duplicate]

How to prove $$\lim_{n \rightarrow \infty}\sqrt[n]{n}=1.$$ I have problem in proving this statement at the beginning my textbook says: Suppose $f_{n}=\sqrt[n]{n}=1+h_{n}$ where does this ...
0
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2answers
29 views

Understanding convergence of fixed point iteration

I was reading some slides explaining the convergence of the fixed point iteration, but honestly I'm not seeing or having an intuitive idea of how fixed-point iteration methods converge. Assuming ...
0
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0answers
26 views

Proving particular case of Abel's series convergence criterion

I need the following to proof a particular case of the Abel's general convergence criterion which goes as follows: Let $\Omega$ be a non-empty subset of $\mathbb{C}$ and A a non-empty subset of ...