Convergence of sequences and different modes of convergence.

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2
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2answers
29 views

If $\sum_{n=1}^\infty a_n$ converges and $\sum_{n=1}^\infty \frac{\sqrt a_n}{n^p}$ diverges, then p $\in$ {?}

Let {$a_n$} be a sequence of non-negative real numbers such that the series $$\sum_{n=1}^\infty a_n$$ is convergent. If p is a real number such that the series $$\sum_{n=1}^\infty \frac{\sqrt ...
3
votes
0answers
24 views

Convergence problem in different norms

We remember the definition of strong convergence of a sequence in $B(H)$ for $H$ a Hilbert space: $a_n\rightarrow a$ iff $\left\|a_nx-ax\right\|\rightarrow0$ for all $x\in H$. In general norm ...
0
votes
3answers
63 views

How to prove $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ converges?

Could you please give me some hint how to prove this statement: If $0<a_n<1$ for each n, then $\sum_{n=1}^\infty \frac{a_n}{1+a_n}$ converges iif $\sum_{n=1}^\infty\frac{a_n}{1-a_n}$ ...
2
votes
1answer
32 views

Two questions on Banach-valued spaces of integrable functions

Let $V$ be a Banach space with dual $V'$ and suppose that $V$ is included into the Hilbert space $H$ so that the inclusion is continuous and dense. Then after identification $H = H'%$ we have $V ...
0
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0answers
15 views

How come (1,1,…,1)T is also a eigenvector associated to eigenvalue 1 for Markov Chains?

As much as I know, for irreducible Markov chains there's a unique eigenvector associated to the eigenvalue 1, and which is the stationary regime π of the power method applied to the Transition Matrix ...
1
vote
4answers
109 views

Does the sequence $\cfrac{n!}{\pi^n}$ converge or diverge and why? [on hold]

The problem states: Use the appropriate limit laws and theorems to determine the limit of the sequence or show that it diverges. $$C_n = \frac{n!}{\pi^n}$$
0
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0answers
4 views

Limite of the distance of the iteration to each set

Let $C_1$, $C_2$ convex and closed sets such that the intersection is noempty. I want to show that the iteration $x^{k+1}=f(x^k)$ generated by the function $f: \mathbb{R^n} \to \mathbb{R^n}$ defined ...
3
votes
1answer
50 views

What is the general limit theorem?

There are simple limit theorems like http://archives.math.utk.edu/visual.calculus/1/limits.18/ But they are just special cases. I am quite sure there is an established general result for them. In ...
1
vote
0answers
40 views

How to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$?

Could you please give me some hint how to prove convergence of $\int_0^1f\left(\sqrt x \right)dx$ when f(x) is continuous for $0<x\le1$ and $\lim_{x\to0^+}x^3f^2(x)=1$ ? I tried the usual way: ...
0
votes
1answer
33 views

Absolute convergence does not implies convergence

Find a space and a series that converges absolutely but it does not converges. It is clear that the space can't complete or Banach.
1
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0answers
28 views

Prove that the limit of the function a sequence is the same as the function of the limit of the sequence [duplicate]

Assume that $\lim_{n\to \infty}a_n=a.$ Suppose that the function $f$ is continuous everywhere including at $a$. Form the sequence $(f(a_n))_{n=1}^{\infty}$. Prove that $\lim_{n\to ...
0
votes
0answers
27 views

Radius of Convergence for non-series

What is the radius of convergence of the following function? \begin{align} f(x) = x - \frac{x^3}{3} \end{align} Is there a way to put this in summation notation?
2
votes
1answer
29 views

Some special tests for convergence.

Please, I need some explanations on some special tests for convergence of a series. For example, the ratio test, comparison test and root test. Ratio test: let the limit as $n$ tends to $\infty$ of ...
1
vote
1answer
67 views

Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}$

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+a_n^2}$$ ...
3
votes
2answers
43 views

Convergence of $\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$

We have a positive series $\displaystyle\sum^\infty_{n=1}a_n$. is the following series converge or diverge ?$$\displaystyle\sum^\infty_{n=1}\frac{a_n}{1+n^2a_n}$$ Suppose ...
1
vote
2answers
32 views

Convergence of the complex sequence $c_{n}=\left ( \frac{1}{\sqrt{2}}(1+i) \right )^{n}$?

I got an exercise to determine the sequence converges or not, $c_{n}=\left ( \frac{1}{\sqrt{2}}(1+i) \right )^{n}\in \mathbb{C}$. I re-wrote it to $$c_{n}=\left |c_{n} \right ...
6
votes
3answers
887 views

Which sequences converge in a cofinite topology and what is their limit?

This is an exercise from an earlier calculus 1 reading at my university: Let $X$ be a space containing infinitely many elements. In the cofinite topology, a set $\Omega$ is open iff $\Omega = ...
0
votes
2answers
60 views

Convergence of $\sum_{n=1}^\infty\frac {n^{n}}{e^nn!}$

Check the convergence of: $\displaystyle\sum_{n=1}^\infty\frac {n^{n}}{e^nn!}$ Using the root test I get: $\displaystyle\lim_{n \to\infty} \dfrac {n}{e\sqrt[n]{n!}}$ now I'm left with showing ...
0
votes
1answer
28 views

I need to show that this sequence is increasing and I'm almost there but I need help on last step.

Let $(1+\frac{1}{n})^n$ be a sequence and $f(x)=(1+\frac{1}{x})^x $ on $[1,inf)$. I need to show that f is non-decreasing by showing that $f'(x)\ge0$. So far I have: Let $g(x)=ln(f(x))$, where $ln$ ...
-1
votes
2answers
49 views

Prove that $r^n/n!$ converges? [duplicate]

I need to show that $r^n/n!$ converges where $n\ge r$. Which is basically showing $\lim_{n\to inf}\frac{r^n}{n!}=0.$. Yotas Trejos told I need to do this Let $N$ be an integer number such that $N> ...
1
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0answers
32 views

family of functions/sequences taken over reals instead of naturals

How does convergence for a sequence and a family of functions change when considering $n$ taken from $\mathbb{R}$ instead of the $\mathbb{N}$? for example, consider mollifiers which are defined as ...
1
vote
2answers
32 views

$\frac {a_{n+1}}{a_n} \le \frac {b_{n+1}}{b_n}$ If $\sum_{n=1}^\infty b_n$ converges then $\sum_{n=1}^\infty a_n$ converges as well [duplicate]

We have two positive series: $\displaystyle\sum_{n=1}^\infty a_n$, $\displaystyle\sum_{n=1}^\infty b_n$ and we know that: $\frac {a_{n+1}}{a_n} \le \frac {b_{n+1}}{b_n}$ (from a certain index). ...
1
vote
1answer
51 views

$\displaystyle\Big(1-\frac{t}{n}\Big)^n$ is strictly increasing for $n>N$ and $t>0$

Show that $\exists N\in\mathbb N$ such that, $\displaystyle\Big(1-\frac{t}{n}\Big)^n$ is strictly increasing for $n>N$ $(n\in\mathbb N, t>0)$ Bernoulli Inequality didn't help me I ...
1
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2answers
47 views

Convergence of $\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}}$

Does the following series converges ? $$\displaystyle\sum^\infty_{n=1}\frac {\sqrt[m]{n!}}{\sqrt[k]{(2n)!}} \ \text{for} \ \ k,m\in \mathbb N$$ I tried the ratio test: $ ...
0
votes
1answer
40 views

A question about improper integral

Could you please give me some hint how to solve this problem: Suppose f(x) continuous in $[0,\infty)$ and for each a,b>0 and c>b $ab \left|\int_0^1 f\left(ax+c \right) dx \right|<1$. Prove ...
0
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0answers
15 views

If power series converges to 0 $\forall$ $x \in (-R,R)$, then $a_n$ is $0$ for all $n$

Suppose that $$\sum\limits_{n=1}^\infty a_{n}x^{n}$$ converges for $x \in (-R,R)$. Show that if $f(x)=0$ for all $x \in (-R,R)$ then $a_n=0$ for all $n$. When I look at this , my guess is ...
0
votes
1answer
22 views

Bartle - integration, monotone convergence theorem

Suppose that $(f_n) \subset M^{+}(X, \mathbb{X})$, that $(f_n)$ converges to $f$, and that $\int f d\mu=\lim \int f_n d\mu < +\infty$. Prove that $$\int_E f d\mu=\lim \int_E f_n d\mu $$ for each ...
1
vote
1answer
75 views

Prove that $r^n/n!$ converges where $n\ge r$ [on hold]

The answer is in the title of the question. I need to show it converges to 0 and $r>0$. I am sorry if this is a bad question, I'm having trouble explaining it. So essentially this Do the ...
0
votes
2answers
41 views

Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence but the convergence is not uniform on $[0,1]$

Let $$ f_n(x) := \begin{cases} 1 &\text{for $x$ in } \left(0, \frac{1}{n}\right)\\ 0 &\text{$x$ elsewhere in } [0,1] \end{cases}. $$ Show that $\{f_n\}_{n=1}^\infty$ is a decreasing sequence ...
2
votes
1answer
38 views

Proving uniform convergence

Prove uniform convergence of function series: $$ \sum_{n=0}^\infty \frac{1}{n^2 + x} \sin \frac{1}{n^2 + x}$$ on $ \Bbb R $ I'm stuck with a problem, because I've proven that is uniformly ...
2
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0answers
30 views

When to Interchange Limit & Integral

I really got stuck in the proof of Cauchy's Integral Formula for higher derivatives: Under what Conditions over a function $f$, we can infer that : $\displaystyle\lim_{h\rightarrow ...
0
votes
1answer
24 views

Pointwise limit,$f$, of the sequence is not bounded

Question: Let $f_n(x) := \frac{nx}{1+nx^2}$ for $x \in A := [0, \infty)$. Show that each $f_n $is bounded on $A$, but the point-wise limit of $f$ of the sequence is not bounded on $A$. Does $(f_n)$ ...
0
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0answers
7 views

Accumulation points of sequences modulo $2 \pi$

Motivated by the question of convergence of $\sum_n \frac{|\cos(\sqrt {n\pi})|}{n}$ I considered first the set of all integers $n$ such that $\cos(\sqrt n\pi)>\alpha$, where $\alpha \in (0,1)$, ...
2
votes
1answer
23 views

Radius of Convergence of Sum of two Series.

Hi all, I know there are similar questions on here, but none deal with the fact of trying to prove that $T \geq min\{R,S\}$. Intuitively this doesn't make sense to me, If you have, ...
5
votes
1answer
49 views

Which properties do still hold for the limit of a sequence of functions?

I think it would be very useful to have a list of properties that are preserved for the limit of a sequence of functions, and was wondering if you could help me making the list more complete. Let ...
1
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1answer
31 views

Subspaces and convergence in weak* topology

I would like to ask some questions regarding convergence in the weak* topology and subspaces. Let $X$ be a normed space with subspace $A \subset X$. Assume $X$ is endowed with the weak* topology. ...
1
vote
1answer
40 views

a question about convergence of sequecce!I have tried cauchy method, but it doesn't work

suppose $a_n>0$,and$\sum_{i=0}^\infty a_i$ is convergent,so we need to prove $\sum_{n=1}^\infty{ {1\over n}(a_n+a_{n+1}+\cdots+a_{2n})}$ is also convergent! I have tried cauchy method, but maybe ...
0
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2answers
30 views

How to find the boundaries of a sequence

If $a(n)=\frac{-1}{n!}$ , how does one find the numerical boundaries of this sequence , rigurously ?
0
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1answer
19 views

Ratio test and radius of convergence

I need to find the radius of convergence for: $$\sum \ln j^3 x^j$$ By the ratio test, I get: $$\displaystyle\frac{\ln (j+1)^3 x^{j+1}}{\ln j^3x^j}$$ However, I'm not sure what happens to the ln ...
1
vote
1answer
31 views

Pointwise convergence of arctan(nx)

Question 6 section 8.1 of Introduction to real analysis by Bartle and Sherbert. Show that lim(Arctan nx) = (pi/2)sgn x for x in R, x>=0. I have a final coming up and I've started doing some of the ...
3
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0answers
31 views

Radius of convergence $\infty$

I need to prove that there is no power series representation for f(x)=|x|. I understand the steps, but I am stuck with one part. Proving by contradiction, we say $\displaystyle \sum_{j=0}^{\infty} ...
1
vote
1answer
23 views

Radius of convergence of series

Find the radius of convergence of this series: $$f(x)= \sum_{j=1}^{\infty} \ \frac{(-1)^{j-1}}{j}(x-1)^j$$ I'm not sure what test to use to get the necessary result. I tried using the root test, but ...
0
votes
1answer
18 views

Question about the use of convergence tests

I just found out about convergence tests , found them on Wikipedia and I've got this question : Are you allowed to use , for example , the ratio test if your sequence is NOT defined as a sum ? Let's ...
1
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1answer
44 views

Applying integral test to prove

$f$ is positive, continuous and decreasing on $[N,\infty)$. It is known that: $$\int\limits_N^\infty f(x) \, dx< \infty \,, \text{ then }, \ \sum_{n=N}^\infty a_n \text{ is convergent}$$ Prove ...
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vote
0answers
41 views

Limit of a factorial sequence

Let $$a(n)=-\frac{1}{n!},$$ where $n\ge0$ . How do I prove the convergence of this sequence ? I am not allowed to use the ratio test for this sequence, since this is not a sum, right? So, how do I go ...
0
votes
0answers
26 views

Find an analytic function [duplicate]

Is it possible to find an analytic expression for a smooth, continuous, single-variable function $y=f(x)$ such that: 1) $f(0) = 0$ 2) $f(x_1) = y_1$ 3) $f(x) > 0, \forall x>0$ 4) ...
0
votes
2answers
21 views

A question about series ratio test

Could you please give me some hint how to deal with this question: Suppose $\left|\frac {a_{n+1}}{a_n}\right|\le c_n$ for each n and $c_n<1$. May we conclude that $\left|\frac ...
1
vote
1answer
38 views

Prove Convergence or Divergence

I just need to prove either convergence or divergence for this. Having some serious trouble and would appreciate all help! $$\sum_{n=1}^{\infty}\frac1{n^{1/3}(1+n^{1/2})}$$
0
votes
0answers
8 views

Range of convergence for Taylor's series gf given that of g and of f

Are the following 2 points correct? Let $D_f$ denote the maximal domain for which the Taylor's series of $f$ converges. 1) If $D_g = \mathbb{R}$, then $f$ converges $\implies gf$ converges. 2) On ...
0
votes
3answers
232 views

Showing that $\lim\limits_{n\to\infty}x_n$ exists, where $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + …+\sqrt{n}}}}$

Let $x_{n} = \sqrt{1 + \sqrt{2 + \sqrt{3 + ...+\sqrt{n}}}}$ a) Show that $x_{n} < x_{n+1}$ b) Show that $x_{n+1}^{2} \leq 1+ \sqrt{2} x_{n}$ Hint : Square $x_{n+1}$ and factor a 2 out of the ...