Convergence of sequences and different modes of convergence.

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2answers
30 views

Find all values of x for which the infinite series $S = \sum_{n=0}^\infty\left(\frac{x^2}{x^2+1}\right)^n $ converges

Find all values of x for which the infinite series $S = \sum_{n=0}^\infty\left(\frac{x^2}{x^2+1}\right)^n $ converges, and express $S$ as a function of $x$. I think the interval of convergence is ...
0
votes
0answers
27 views

How to prove this continuous martingale converges?

Suppose $B = (B_t, t \geq 0)$ is standard Brownian motion. Let $M^\lambda_t := \exp(\lambda B_t - \frac{\lambda^2 t}{2})$ (I have previously shown that this is a martingale). How do I prove that $$ ...
0
votes
3answers
63 views

Show whether this trigonometry series converges

$\displaystyle\sum_{n=1}^{\infty}\sin\left(\frac{3n}{1+3^n}\right)$ Kinda obvious that it converges, but how do I prove it mathematically?
0
votes
1answer
14 views

Equivalents definition of linear convergence

Suppose that the sequence $\{x_n\}$ converges to $0$. I want to prove that these definitions are equivalent: a) We say that $\{x_n\}$ converges linearly to $0$, if there exists a number $q \in (0, ...
0
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2answers
46 views

Definition of limit convergence

I want to show that the sequence represented by .9, .99, .999, ... converges to 1 but I'm unsure how to go about picking an epsilon such that it works
9
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2answers
118 views

If $\lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = 0$, can $\sum_0^\infty a_n$ be rational?

If a nonzero sequence of rationals $$a_0, a_1 \dots a_n$$ "decays fast" in the sense that $\lim_{n \rightarrow \infty} a_{n+1}/a_n = 0$, can the series converge to a rational number? That is, can ...
0
votes
0answers
21 views

Examine convergence and (almost) uniform convergence of $f _{n} = n\left[ f\left( x + \frac{1}{n} \right) - f\left( x\right) \right] $

Show almost uniform confergence of: $ f _{n} = n\left[ f\left( x + \frac{1}{n} \right) - f\left( x\right) \right] $ I've noticed that : $f' _{n} = \frac{ f\left( x + \frac{1}{n} \right) - ...
1
vote
2answers
50 views

Show that a power series is analytic inside its radius of convergence

Let $f(z)=\sum_{k=0}^\infty a_k(z-z_0)^k$ with radius of convergence $R$ then $f$ is analytic on the open disk around $z_0$ with radius $R$. What I was thinking about is an approach based on this ...
6
votes
2answers
46 views

Nested logarithm series

While working on problems from Spivak's Calculus, I came on one asking for the convergence/divergence of the series $$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}.$$ This is a straightforward ...
0
votes
0answers
21 views

Radius of convergence of $f(z)?$

Let $p(x)$ be a polynomial of the real variable $x$ of degree $k\geq 1$ .Consider the power series $$f(z)=\sum_{n=0}^{\infty}p(n)z^n$$ where z is a complex variable .Then the radius of convergence of ...
0
votes
1answer
14 views

ratio test is inconclusive

I have a series $\sum_{k=1}^{\infty} \frac{cos(k)}{2*(k^3)-k}$ Using ratio test, I got $={\sum_{k=1}^{\infty} \frac{cos(k+1)}{k*(2*(k+1)^2-1)}}/{\frac{cos(k)}{k*(2*(k^2)-1)}} = ...
-2
votes
0answers
29 views

Check convergence $ \sum_{}^{} \frac{n ^{2}x ^{2} }{e ^{n ^{2}\left| x\right| } } $

Check the pointwise, uniform, and almost uniform convergence of: $ \sum_{}^{} \frac{n ^{2}x ^{2} }{e ^{n ^{2}\left| x\right| } } $
3
votes
4answers
89 views

Convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$

I want to test the convergence of the integral $\int_0^\infty e^{-x^3} \mathop{\mathrm{d}x}$. There are some parts of the solution which does not make sense to me, I'm hoping that someone can explain ...
0
votes
1answer
52 views

Examine convergence and (almost) uniform convergence of $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$.

How to examine convergence, almost uniform convergence and uniform convergence of series $\sum_{n=1}^{+ \infty} \frac{n^2 x^2}{e^{n^2 |x|}}$ for $x \in \mathbb{R}$?
0
votes
0answers
12 views

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 a distribution over the whole $\mathbb{Z}^+$, where $p(k) = \gamma_k$. We are interested in approximating $p(v)$ ...
-1
votes
0answers
7 views

An examination of rates of convergence of the series

I check the website they all talked about rate of convergence of sequence. Can anyone gives me an example for rates of convergence of the series?
1
vote
1answer
18 views

Sum of two random variables converging with different modes [closed]

Is it true that if X_n converges in distribution to X; Y_n converges in probability to Y; X_n, Y_n, X and Y are real-valued random variables defined on the same probability space, then X_n + Y_n ...
-1
votes
3answers
47 views

Determine whether the series converges

$\sum\limits_{n=1}^{\infty} \dfrac{1}{n^{1+1/n}} $ I believe it diverges. But I am having trouble comparing it to another series which also diverges and whose terms are less than the original ...
0
votes
1answer
47 views

Given that an integral containing $|x_{n}|$ is bounded by a constant, show that $x_{n}\to x$ in $L^{1}(0,\infty)$

This question is related to another question I asked earlier. For reference, this is the relevant part of that question: Let the sequence of continuous functions $\mathbf{\{x_{n}(t) ...
-2
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0answers
23 views

explain why there is an observed rate of convergence

Using your knowledge and theorems explain why there is an observed rate of convergence when using the composite simpsons rule?
0
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0answers
20 views

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$

$d:\mathbb N \times \mathbb N \to \mathbb R \ d(m,n)=0:m=n ; d(m,n)=1-\frac{1}{m+n}:m \neq n. $ Question convergence of sequences $(x_n)(y_n)(z_n)$ and prove that $B_n=\{m \in \mathbb N : d(m,n)\leq ...
0
votes
1answer
17 views

Criteria for convergence of power series

Given the power series: $\; \sum_{i=0}^{\infty}a_nz^n \;$ Proof that if there exist $s,M \in \mathbb R $ such that $|a_n|s^n \le M$ then the power series converges for every $|z|\lt s$ Can someone ...
1
vote
1answer
16 views

Uniform convergence of function series, convergence on intervals

I I'm interested in, if it is true: If series $\sum f_k$ is uniformly convergence on every interval $[n; n+10]$, where $n \in \mathbb{Z}$, then $\sum f_k$ is uniformly convergence on $\mathbb{R}$. ...
1
vote
1answer
32 views

Evaluate $\lim_{x\to 0}(x^2( 1+2+…+\left[\frac{1}{|x|}\right]))$, where $[a]$ is the largest integer not greater than $a$, a is real

My question is that Evaluate $\lim_{x\to 0}(x^2( 1+2+...........+\left[\frac{1}{|x|}\right]))$ For any real number a, [a] is the largest integer not greater than a. My approach :-The series can be ...
1
vote
1answer
30 views

How to test convergence for $\sum^{\infty}_{n} \frac{1}{(\ln{n})^3}$?

My try: using basic comparison. Since $\ln{n}\lt n^\frac13$ for large $n$, $\frac{1}{\ln(n)}\gt \frac{1}{n^\frac13}$. Since $\sum^{\infty}_{n} \frac{1}{(n^\frac13)^3}$ diverges, so $\sum^{\infty}_{n} ...
2
votes
3answers
42 views

Study the convergence of the following series $\sum\limits_{n=1}^\infty \frac{n}{n^2+3} \sin(\frac{1}{\sqrt{n+2}}) $

I have to study the convergence of the following series: $\sum\limits_{n=1}^\infty \frac{n}{n^2+3} \sin(\frac{1}{\sqrt{n+2}}) $ Is a positive series, so I should divide for $\frac{1}{\sqrt{n+2}}$ ...
0
votes
0answers
15 views

Fubinu's principle on series

If one double series is absolutely convergent then $$\sum_{j\geqslant 0}\sum_{i \geqslant 0}a_{ij}=\sum_{i \geqslant 0}\sum_{j \geqslant 0}a_{ij}$$ I looked around and (probably not very good) and ...
0
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0answers
27 views

Convergence of Algorithm in optimization

$\bf[8.54]$ Let $f:\Bbb R^n\to\Bbb R$ be differentiable. Consider the following procedure for minimizing $f$: $\qquad$ Initialization Step $\quad$ Choose a termination scalar ...
0
votes
1answer
41 views

Prove a.s. convergence of random variables.

I need to prove this: Assume that you have a probability space $(\Omega, \mathcal{F},P)$, $X_t$ is a stochastic process which is jointly measurable with respect to $\mathcal{B}(\mathbb{R})\times ...
1
vote
1answer
33 views

Limit with logarithm: $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n}$

What is the limit $\lim_{n \to \infty} \frac{n^\alpha}{\ln^\beta n }$ (ln=natural logarithm) for alfa real and less than zero? I found out it is zero for $\beta\ge0$, since then you can use the ...
0
votes
1answer
32 views

Let $(x_\alpha)_{\alpha\in J}$ be a net in a topological space $X$. Show that if $J$ is a finite set then the net converges? …

Let $(x_\alpha)_{\alpha\in J}$ be a net in a topological space $X$. Show that if $J$ is a finite set then the net converges? ... Why does it have to converge? I don't understand why with nets this is ...
1
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3answers
61 views

Proof $\sum \frac{1}{n^a}$ is convergent for a > 1

I get to the fact that $\sum_{k=1}^n \frac{1}{k^a}$ < $\frac{1}{a-1} - \frac{1}{(a-1)(n+1)^{a-1}} - \frac{1}{(n+1)^a} + 1$ and hence $\sum_{k=1}^n \frac{1}{k^a}$ is bounded. How to deduce $\sum ...
0
votes
1answer
32 views

Show the sequence [fn]= 1+(1/1!)+(1/2!)…+(1/n!) is increasing and bounded above by 3. [duplicate]

This is part of a question. In the end we are trying to show the sequence up above converges to e. I need to use math induction.
0
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2answers
31 views

Convergence of $(-1)^n/\sqrt{n}$

So, I don't know how correctly show that, $c(n) := \left(\frac{(-1)^n}{\sqrt{n}}\right)\to 0\qquad\text{ as}\quad n\to\infty.$ Should I do this with limes or by $\forall \epsilon > 0, ...
0
votes
2answers
49 views

A tricky question involving convergence of series

I'm struggling to find a title for this. Hopefully it can be edited subsequently. Suppose $\sum_k s_k(x)$ converges to $S(x)$ Now suppose each $s_k(x)$ has a sequence $t_k^N(x)$ approaching it (as ...
5
votes
2answers
46 views

Does $\sum^{\infty}_{n=1}\frac{\ln{n}}{n^{1.1}}$ converge or diverge?

Does $$\sum^{\infty}_{n=1}\frac{\ln{n}}{n^{1.1}}$$ converge or diverge? I think the basic comparison works but I have a hard time finding a comparer. Could someone suggest one?
0
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0answers
21 views

How to test the convergence of the following sereis?

$\sum^{\infty}_{n=1}\sin{\frac{1}{n}}$: The only test I can think of for this one is basic comparison ($\sin{\frac{1}{n}}\le\frac{1}{n}$). But $\frac{1}{n}$ diverges. ...
0
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1answer
14 views

Show a certain sequence is bounded

Let $s$ satisfy: $0<s<1$. Show that $ns^{n-1}$ is bounded $\forall n\ge1$. Thoughts so far: If we can treat the $ns^{n-1}$ as a sequence and show it is convergent, it is easy to show it is ...
1
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3answers
41 views

How to prove that the improper integral $\int^{\infty}_{1}\frac{\ln{x}}{1+x^2}dx$ is convergent?

How to prove that the improper integral $\int^{\infty}_{1}\frac{\ln{x}}{1+x^2}dx$ is convergent? I tried both basic and limit comparison but I couldn't find a proper measure. i.e. ...
0
votes
4answers
38 views

$\sum_{n=1}^{\infty} \frac{\sin(1/n)}{\sqrt n}$ radio test convergence or divergence of series

I have to check if this converges or diverges using the ratio test which is the $\lim_{n\to\infty} | a_{n+1} / a_{n} | $. The problem is : $$\sum_{n=1}^{\infty} \frac{\sin\frac1n}{\sqrt n}.$$ So ...
2
votes
2answers
30 views

How to test the convergence of $\sum^{\infty}_{n=1} \frac{3^n(n-1)(-1)^n}{n^3}$?

So $\sum^{\infty}_{n=1} \frac{3^n(n-1)(-1)^n}{n^3}=\sum^{\infty}_{n=1} \frac{(-3)^n(n-1)}{n^3}$. Then I fail to continue. Is there a way to simplify it?
1
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1answer
32 views

A topology on a set $X$ is exactly determined by specifying when, for $x\in X$ and sequences $x_n \in X, x_n \to x, $ as $n \to \infty$.

A topology on a set $X$ is exactly determined by specifying when, for $x\in X$ and sequences $x_n \in X, x_n \to x, $ as $n \to \infty$. This is a statement from my functional analysis notes, but I ...
3
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1answer
27 views

Does this trigonometry series $\displaystyle\sum_{n=1}^{\infty}\frac{\sin3n}{1+3^n}$ converges?

Is there any tests or method by which I could test the convergence of the following series: $$\sum_{n=1}^{\infty}\frac{\sin3n}{1+3^n}$$
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0answers
17 views

Specific example where uniform convergence of complex series does not imply normal comvergence necessarily [duplicate]

I posted this question earlier, but nobody could help me: Example where uniform convergence does not imply normal convergence I was wondering if some more people could take a look. Thank you so ...
4
votes
1answer
33 views

Reordering a conditionally convergent series

I have the series $$\sum_{n=1}^\infty(-1)^{n+1}\frac{1}{n}=\ln(2),$$ and I want to reorder it to $$\sum_{n=1}^\infty\frac{8n-3}{2n(4n-3)(4n-1)}.$$ If we write the terms of the first series we get ...
2
votes
0answers
30 views

How to prove convergence for the following?

Let $X_m$ be distributed as follows: $$\mathbb P(X_m=\pm m)=\frac{1}{2m^\beta}, \mathbb P(X_m=0)=\frac{m^\beta-1}{m^\beta}.$$ Let be $S_n=\sum_{m=1}^{n}X_m$, then I. If $\beta>1$, then ...
0
votes
0answers
30 views

If $f_n$ is in $L^1$ and $f_n$ converges to $f$, is $f$ in $L^1$?

This question might come off as basic to most of you, but this isn't basic to me. If $f_n$ is in $L^1$ and $f_n$ converges to $f$ in $L^1$ ($||f_n-f||_1 \to 0$ as $n\to \infty$), does it necessarily ...
0
votes
0answers
7 views

Rate of expected value of $\mathcal{O}_p$

This is certainly very basic but what is the rate of the expected value of a random variable that is bounded in probability. For example, let $X_n = \mathcal{O}_p (a_n)$ is it true that $\mathbb{E} ...
3
votes
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 $x$?