Convergence of sequences and different modes of convergence.

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Weierstrass M-test help

I am supposed to use M-test on this one $$\sum \frac {n\ln (1+nx)}{x^n}$$ on $$1<x< \infty$$ But I face problems finding an appropriate $M_n$, thanks for help
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38 views

Can someone help me to choose the best answer to prove the series converges

Can someone please help me to choose the better answer. Question : Solution 2)
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1answer
43 views

which convergence test is better?

I need to check this one for absolute convergence $$\sum^{\infty}_1 \frac {(-1)^n(n+4)}{(n^2+1)^{1/4}(2+\sqrt{n^2+3})}$$ But I am not sure which method to use, it fails with Root or Ratio tests.
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1answer
17 views

Series convergence problem for positive x

Having the following sequence: $\sum\limits_{n=1}^\infty \frac{n!}{\left(x+1\right)\left(x+2\right) \ldots \left(x+n\right)}$ $x>0$ how to investigate its convergence? Which criteria should be ...
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22 views

If I'm asked to prove {1/n} from n=1 to infiniti converges to 0, can I assume the euclidean metric?

Using the definition of limit If I'm asked to prove {1/n} from n=1 to infiniti converges to 0, can I assume the euclidean metric?
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28 views

Expected value of the negative portion of sum of poisson random variables

Setting: Defn: for every $x \in \mathbb{R}$ define its negative part by $x^{-} = -x$ if $x \leq 0$, and $x^{-} = 0$ if $x > 0$ Let $\{X_j, j \ge 1\}$, $X_j \overset{d}{\sim} Poisson(1) = \Pr\{X = ...
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1answer
20 views

Convergence of a series with trigonometric functions

I have two sequences as follows: $\sum\limits_{n=1}^\infty \frac{\sin n\phi}{n}$ $\sum\limits_{n=1}^\infty \frac{\cos n\phi}{n}$ How to investigate convergence of those two? What criteria should i ...
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1answer
27 views

Estimating convergence with Liapunov function

we have the system: $x' = 2y(z-1) - x^3, y' = -x(z-1) -y^3, z' = - z^3$ and consider its equilibrium at $\vec{o}$. We want to prove that for every non-trivial solution there exists two positive ...
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1answer
11 views

fast way to find order of a (semi) geometric series

Is there a fast way to find whether the order of the following is $O(T^2)$ or $O(T)$? I've been trying to find the exact thing by using the geometric series multiple time, but it is so lengthy and ...
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1answer
10 views

Ratio test for testing the convergence of infinite series.

I am wondering does it make a difference for the ratio test if $a_n$ is a series and we take $lim |\frac{a_n}{a_{n+1}}|$ will it be the same as taking the limit $lim |\frac{a_{n+1}}{a_n}|$
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8 views

Convergence of recursive application of finite-difference operator to $C^{\infty}$ functions

Let $f\colon \mathbb{R}\to \mathbb{R}$ be an arbitrary smooth function (whose extension to a complex differentiable function is entire, if it matters). Let $\mathbf{D}_{h}$ be a finite difference ...
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25 views

Is it always true that if there is only one subsequential limit point of a sequence, then the sequence must converge to it?

If x is the only subsequential limit point of a sequence, is it always true that the sequence converges to x?
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15 views

How do I calculate the following radius of convergence?

Original equation: To determine de radius of convergence. I use R = 1/L. Where R is the radius and L is the limit. The answer is supposed to be limit L = 0. So R = 1/0 = ∞ So this limit is ...
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1answer
56 views

Convergence of $a_n = (1+i)^n+(1-i)^n$

I am asking a question related to Is there a formula for $(1+i)^n+(1-i)^n$? I am looking on the exact same term, just as a sequence, so i want to find out: Is $a_n = (1+i)^n+(1-i)^n$ convergent or ...
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1answer
36 views

Some help to prove the series converges

Can someone please correct my answer I am not sure about it
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1answer
19 views

Example of convergence in probability to a non-degenerate rv

Suppose the sequence of random variables, X$_n$, converges in probability to another random variable X. The condition requires that for any arbitrary distance, $\epsilon$, the probability that the ...
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2answers
45 views

Determine whether the series $\sum\limits_{n=1}^\infty \frac{n!+n}{(n+2)!}$ is convergent or divergent.

Determine whether the series $$\sum\limits_{n=1}^\infty \frac{n!+n}{(n+2)!}$$ is convergent or divergent. Wolfram Alpha says that "By the comparison test, the series converges" but I can't ...
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1answer
36 views

Convergence of truncation in $L^{p}$

If you have a truncation $T_{k}u$ defined as: $$ T_{k}u := \begin{cases} u,& \text{ if }~ |u(x)| \leq 1\\ k\frac{u}{|u(x)|}, & \text{ if }~|u(x)| > k \end{cases} $$ If you consider ...
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1answer
32 views

Does weak convergence in $L^{q}$ imply weak convergence in $L^{p}$

Assume we have $u_{k} \rightharpoonup u$ in $L^{q}(\Omega)$, does it then follow that $u_{k} \rightharpoonup u$ in $L^{p}(\Omega)$, given that $q > p$ and $\Omega \subset \mathbb{R}^{n}$ is ...
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90 views

Determine the convergence of infinite series $1-1+\frac{1}{2}-\frac{1}{2}+\frac{1}{3}-\frac{1}{3} +\cdots$

How shall I determine whether the series $$1-1+ \left(\frac{1}{2}\right)-\left(\frac{1}{2}\right)+\left(\frac{1}{3}\right)-\left(\frac{1}{3}\right)+\cdots$$ is convergent or divergent? Please help?
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31 views

Absolute convergence2

I have to test the series for absolute and conditional convergence $\sum_{n=2}^{\infty}$ $\frac{(-1)^{n-1}}{n^2+(-1)^n}$ $Notes :$ For absolute convergence def. I have $\vert ...
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1answer
41 views

Showing that the indicator/characteristic function is not a regulated function

I want to show that the indicator function (aka. the characteristic function) is not a regulated function. \begin{align} \chi : \begin{cases}[a,b] & \longrightarrow \mathbb{R} \\ x & ...
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2answers
108 views

Deciding whether series containing $a_n$ are convergent knowing that $\lim_{n\rightarrow\infty}\frac{\frac{(-1)^n}{\sqrt{n}}}{a_n}=1$

We know the following thing about sequence ${a_n}$: $$\lim_{n\rightarrow\infty}\frac{\frac{(-1)^n}{\sqrt{n}}}{a_n}=1$$ And now the problem asks us whether it's true for every such $a_n$ that: ...
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26 views

convergence test

I have to test if the series is absolute convergence and conditional convergence $\sum_{n=1}^{\infty}$$\frac{(-1)^{n-1}n}{(n+1)^2}$ This what I have so far: Im going to test for absolute ...
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2answers
26 views

Convergence of sequences in $\mathbb{C}$

I am currently getting into the field of complex numbers, with the imaginary unit $i^2 = -1$ and stuff. At the moment i am looking onto a few sequences in $\mathbb{C}$, regarding convergence. I have ...
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3answers
51 views

Find the sum of the series with unordered powers of $3$

Consider the following series: $$\sum_{n=1}^{\infty} a_n = 1/3+1+1/3^3+1/3^2+1/3^5+1/3^4+1/3^7+1/3^6 +\dots$$ Determine if it converges, and find the sum. Here is what I got: a) ...
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1answer
16 views

Trouble with Example for Convergence in Distribution

I am a bit confused by an example used to illustrate the concept of "convergence in distribution" Intuitively, this makes sense, since if we choose a large number of points from the distribution ...
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1answer
30 views

Markov chain modes of convergence

This is continuation of the question stated here. Let $\left( {{X_\alpha }:\alpha \in A} \right)$ be a finite space Markov chain (discrete or continuous), consisting of only transient and absorbing ...
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1answer
29 views

Estimate the speed of convergence to the stationary distribution for a ergodic Markov process

I have encountered a Markov process with following transition matrix $P= \begin{bmatrix} 0.6 & 0.4 \\ 0.2 &0.8\end{bmatrix} $. This is an ergodic Markov matrix since all the elements are ...
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81 views

Evaluating $\sum_{n=1}^{\infty}\frac{6}{n(n+1)(n+2)}$

Show that series $$\sum_{n=1}^{\infty}\frac{6}{n(n+1)(n+2)}$$ converges by simplifying its sequence of partial sums and find its sum. I don't have much detail but this all I have: ...
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3answers
45 views

Determine whether or not the following sequence converges. If so, find the limit.

Im not sure if I am evaluating the convergence of the following sequence correctly, and I am unsure of how to determine the limit. All help is greatly appreciated. $a_n = {1 \over ...
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50 views

Is the series $\sum_{n=1}^{\infty}$ $(\sqrt[n]{2n^2}-1)^n$ convergent or divergent?

The problem is to determine whether the series $\sum_{n=1}^{\infty}$ $(\sqrt[n]{2n^2}-1)^n$ converges or diverges. Proof of the convergence os the series: Let $U_n = (\sqrt[n]{2n^2}-1)^n$. As ...
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19 views

Proving according to Cauchy criterion that an alternating harmonic series with factorial denominator converges.

I need to prove according to Cauchy's criterion that the sequence: $a_n = 1 - \frac{1}{2!} + \frac{1}{3!} - ......... + \frac{1}{(2n-1)!} - \frac{1}{(2n)!} $ converges. but I got stuck with the ...
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65 views

$\sum_{n=1}^{+ \infty} a_{n} \frac{1-e^{-n^{2}t}}{n} = 0$ implies $a_{n} = 0$?

Say that for the coefficients $a_{n} \in \mathbb{R}$, we have that $$ \sum_{n=1}^{+ \infty} a_{n} \frac{1-e^{-n^{2}t}}{n} = 0$$ for every $t > 0$. Does this imply that $a_{n} = 0$ for every $n \in ...
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32 views

Is this statement about limits true.

I am wondering if this is true: If $\{a_n^k\}_{n=1}^\infty\rightarrow a^k$, and $\{a^k\}_{k=1}^\infty\rightarrow a$, does then $\{a_n^n\}_{n=1}^\infty\rightarrow a$? I tried proving it but I got ...
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10 views

Solution Verification for a Convergence Question

The Question Give an example of a pair of series $\sum a_n$ and $\sum b_n$ with positive terms where the limit as n goes to infinity of $\frac{a_n}{b_n} = 0$ and $b_n$ diverges and $a_n$ converges. ...
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33 views

Divergence test for the series $\sum_{n=1}^{\infty} (\sqrt[n]{2n^2}-1)^n$

Determine the convergence of $$\sum_{n=1}^{\infty} (\sqrt[n]{2n^2}-1)^n$$ I have no idea what test I should use any hint? I think is divergent, I was thinking using comparison test but don't ...
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1answer
30 views

Does the following alternating series converge or diverge?

I have the following series that I have to check for convergence or divergence: $\sum\limits_{n=1}^{\infty} \frac{sin(n+ 1/2)\pi} {1 + \sqrt{n}}$ I know that it is an alternating series therefore I ...
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1answer
39 views

Convergence test for the series $\sum_{n=1}^{\infty} \frac{\sqrt{n^2+1}-n}{\sqrt{n}}$

Determine convergence of the series $$\sum_{n=1}^{\infty} \frac{\sqrt{n^2+1}-n}{\sqrt{n}}$$ My proof: using comparison test I have $$\frac{\sqrt{n^2+1}-n}{\sqrt{n}} = \sqrt{n+\frac{1}{n}} ...
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35 views

Uniform Continuity and Limits

Let $f\colon (0, 1]\to\mathbb{R}$ be a function which is uniformly continuous. (i) Show that if $\{x_n\}$ is a sequence with $x_n > 0$ and $\lim_{n\to∞} x_n = 0$, then $\{f(x_n)\}$ is convergent. ...
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44 views

Sum of exponential functions involving powers of two

I came across a weird series with exponential functions and powers of two: $$\sum_{k=0}^{\infty} \left(1 - e^{-2^{-k}z} \right), z \in \mathbb R_+$$ and have no idea how to solve this (if there even ...
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40 views

for which $n$ does $\int_0^1 sin(x)^n|ln(x)|^2dx$ converges?

I wonder for which real number $n$ the integral $\int_0^1 sin(x)^n|ln(x)|^2dx$ converges. If $n\geq 0$ i think the answer is yes, since $|sin(x)^n|ln(x)|^2|\leq |ln(x)|^2$ and $\int_0^1 |ln(x)|^2dx$ ...
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1answer
26 views

Is the following series is converging, diverging, or?

This is a fairly basic homework question so I'm not expecting a full answer but some hints would definitely be much appreciated. Establish whether this series diverges, converges, or whether there is ...
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2answers
33 views

Limit of $P(X_n > a_n)$ where $X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$ and $a_n\xrightarrow[n \to \infty]{} \infty$

I've been working on following problem and could need some help. Let $X_n$ be a sequence of RV with $$X_n \xrightarrow[n \to \infty]{d} X \sim{N(\mu,\sigma^2)}$$ for some $\mu \in \mathbb{R}$ and ...
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17 views

Rate of convergence of the difference of two exponentials

I would like to find the convergence rate of the following function: $$f(x) = |e^{-ax}-e^{-bx}|,$$ with $a,b>0$ and $x\to+\infty$. By finding the convergence rate, I mean finding the largest ...
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4answers
48 views

Convergent sequence? [duplicate]

Why does the $\lim \sum{n^{(1/n)}-1}$ diverge as $n\rightarrow \infty$? I suspected it would converge as $\lim {n^{(1/n)}}=1$ as $n\rightarrow \infty$ but computation show otherwise. So, now I am ...
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42 views

Is there a better solution (?)

There are two sequences $ a_n $ and $ b_n $, which fulfill 3 certain properties: The series of one of them converges, while the other diverges. $ \frac {a_n} {b_n} $ = 1, if n tends to infinity. ...
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3answers
35 views

Prove convergence for p>1

Prove the series is convergent for p>1: $\sum _{k=1}^{\infty } \frac{1}{(n)(ln(n)^p)}$ The ratio test is inconclusive. I assume a comparison test is involved but am not sure how to make that happen. ...
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20 views

$L^2$ limit of Gaussian random variables

Let $X_n$ be a sequence of Gaussian random variables defined on the same probability space. The statement is that if $X_n$ converges to some random variable $X$ in $L^2$-sense, then $X$ is also ...
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1answer
29 views

Power series with $x^{4n}$

I'm new to this Forum. I do not find an approach to solve the following problem (from the book "Herbert Wallner, Aufgabensammlung Mathematik Band 1", so this is not a homework question): For which ...