What is Convergence: Definition and 1000 Discussions
CONvergence is an annual multi-genre fan convention. This all-volunteer, fan-run convention is primarily for enthusiasts of Science Fiction and Fantasy in all media. Their motto is "where science fiction and reality meet". It is one of the most-attended conventions of its kind in North America, with approximately 6,000 paid members. The 2019 convention was held across four days at the Hyatt Regency Minneapolis in Minneapolis, Minnesota.
HI guys,this is my first programming experience , i have developed an MATLAB code for steady state heat conduction equation , on governing equation
dt2 /dx2 + dt2/dy2 = -Q(x,y)
i have solved this equation with finite difference method, As far as i know if we increase the mesh size it leads...
Homework Statement
I've been given a question that makes use of 5^(n)*sin(pi*n!)
The question merely asks if the sequence converges, and if so, to determine its limit. Am I right in assuming that this does converge, under the definition, but does so as n-> - infinity?
So basically, I...
Homework Statement
Show that if given \mathbf{x}_0, and a matrix R with spectral radius \rho(R)\geq 1, there exist iterations of the form,
\mathbf{x}_{n+1}=R\mathbf{x}_0+\mathbf{c}
which do not converge.
The Attempt at a Solution
Let \mathbf{x}_0 be given, and let...
We work on project about mobile cellular networks. In a part of the project, we faced the problem about the proof of convergence of ...(Complete description is Available in attachment.Please download it)
Please help me.It is very important!
1.
What if absolute convergence test gives the result of 'inconclusive' for a given power series?
We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...
my final is tomorrow and my instructor gave a list of questions that will be similar to the ones on the final exam and i want to see how they should be done properly. I've been working on other problems but i can't get past these ones. thanks
determine convergence/divergence...
find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence.
i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?
find the interval of convergence of the series $\sum_{x=1}^{\infty} \frac{6x^n}{\sqrt[5]{n}}$find the radius of convergence of the series $\sum_{n=1}^{\infty} \frac{8^nx^n}{(n+5)^2}$
I'm currently reading Tolstov's "Fourier Series" and in page 58 he talks about a criterion for the convergence of a Fourier series. Tolstov States:
" If for every continuous function F(x) on [a,b] and any number ε>0 there exists a linear combination
σ_n(x)=γ_0ψ_0+γ_1ψ_1+...+γ_nψ_n for which...
Hi. First off, sorry for the not so descriptive title. If one of you finds a better tilte I will edit it.
We have the equation
\begin{equation}
\partial_{xx}\phi = -\phi + \phi^{3} + \epsilon \left(1- \phi^{2}\right)
\end{equation}
We will look for solutions satisfying...
Definition/Summary
In what follows, we will work in a normed space (X,\|~\|).
A series is, by definition, two sequences (u_n)_n and (s_n)_n such that s_n=\sum_{k=0}^n{u_k} for every n.
We call the elements u_n the terms of the series. The elements s_n are called the partial sums. We will...
Hello.
How do I find the radius of convergence for this problem?
##\alpha## is a real number that is not 0.
$$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$
I understand that we can use the ratio test to find R. And by using ratio test, I got R=1. But in the...
Hello.
How do I find the radius of convergence for this problem?
$\alpha$ is a real number that is not 0.
$$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$
Hello.
I am stuck on this question. I'd appreciate if anyone could help me on how to do this.
The question:
Expand the following into maclaurin series and find its radius of convergence.
$$\frac{2-z}{(1-z)^2}$$
I know that we can use geometric series as geometric series is generally...
on page 4, example 9 in this link, http://www.personal.psu.edu/auw4/M401-notes1.pdf, they show a sequence of functions is not uniformly convergent. To show this, you need to show that for some epsilon, there is no 'universal' N.
But they didn't pick a particular value of $z$, they chose...
Hello.
I need explanation on why the answer for this problem is R = ∞.
Here's the question and the solution.
Expand the function into maclaurin series and find the radius of convergence.
$zsin(z^2)$
Solution:
$$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$...
Hello.
I need explanation on why the answer for this problem is $R=\infty$.
Here's the question and the solution.
Expand the function into maclaurin series and find the radius of convergence.
$zsin(z^2)$
Solution:
$$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$
Divide...
Just a few quick questions this time:
I'm doubting the first one mostly, because when I used the integral test to evaluate it: I ended up getting (-1/x)(lnx +1) from 2 to infinity, which gave me an odd expression: (-1/infinity)(infinity +1 -ln2 -1). I'm assuming this means it is convergent...
Hey guys, I have a couple more questions.
For the first one, taking the limit to infinity obviously equals 0 so it should be convergent, right?
Also, for the second one, the limit as n approaches infinity for gives me indeterminate form, so I took the derivative which just gave me ln(n)...
Hey guys,
I have a few quick questions for the problem set I'm working on at the moment:
I'm highly doubtful of my answer for c. I used the roots test instead of the ratio test, which gives 1/n, which I took the limit of to get an interval of [-∞ , ∞]
As for a and b, I got [-5,5] and (-∞, ∞)...
Hey guys,
I have a few more questions for the problem set I'm working on at the moment:
I'm unsure about b in particular. I compared the series to 1/(n^3/2), which makes it absolutely convergent by the p-test and comparison test. Do I still have to perform any other tests to confirm absolute...
Hey guys,
I have a few quick questions for the problem set I'm working on at the moment:
I'm mostly unsure of my response for b. For a, I just split the series into two parts and added 6+3 to get 9, and thus the series is convergent. For c, I got 3/5 after taking the limit, which is...
This thread is only for question 5.
As for number 5 part a, after tediously expanding the partial fraction expression, I ended up getting c=1, d=0, b=1, and c=1, ultimately resulting in: ln(x) - (1/x^2) + c. I really don't think this looks right.
As for 5b, I obtained b=-1, c=-1, a=2, and...
For a,b, and c respectively, I got divergent (to -infinity), convergent (to π/6), and divergent (to infinity, since the first part's sum is 1/3, but lim negative infinity gives infinity, thus the summation of the two integrals gives a divergent integral). I'm sure these are right, but I'd...
Hello,
I'm doubting a couple of my answers for these questions. Some of them seem relatively simple, but there are slight nuances that I'm not sure of.
This thread is only for question 4.
For 4a, I just used the (a^2) - (x^2) => x=asin(Ø) rule and substituted 3sin(Ø) for x. I ended up...
(x-1)-\frac{(x-1)^2}{2!}+\frac{(x-1)^3}{3!}-\frac{(x-1)^4}{4!}+ ∙ ∙ ∙
well this looks like an alternating-series, the question is: at what value(s) of x does this
converge.
one observation is that if x=0 then all terms are 0 so there is no convergence, also I presume you can rewrite this as...
Homework Statement
Does the following series converge or diverge? If it converges, does it converge absolutely or conditionally?
\sum^{\infty}_{1}(-1)^{n+1}*(1-n^{1/n})
Homework Equations
Alternating series test
The Attempt at a Solution
I started out by taking the limit of ##a_n...
Homework Statement
Let ##X## be a topological space. Let ##A_1 \supseteq A_2 \supseteq A_3...## be a sequence of closed subsets of ##X##. Suppose that ##a_i \in Ai## for all ##i## and that ##a_i \rightarrow b##. Prove that ##b \in \cap A_i##.
Homework Equations
The Attempt at a Solution...
Homework Statement
For which integer values of p does the following series converge:
\sum_{n=|p|}^{∞}{2^{pn} (n+p)! \over(n+p)^n}
Homework Equations
The Attempt at a Solution
I'm trying to apply the generalised ratio test but get down to this stage where I'm not sure what...
Homework Statement
Does the following series converge or diverge? ##∑\frac{n^5}{n^n}## (as n begins from 1 and approaches infinity)
Homework Equations
Ratio test?
The Attempt at a Solution
For your reference, thus far I have learned about the geometric series, the limit test...
Homework Statement
For 0<q<∞, and x rational, for what x values does the series converge?
\sum_{n=0}^{∞} q^{1/n} x^nThe Attempt at a Solution
I don't know which method works best for this
Homework Statement
Given a non-negative sequence \{a_{n}\}_{n=1}^{\infty}. Proove that the serie \Sigma_{n=1}^{\infty}a_{n} converge if and only if \Sigma_{n=1}^{\infty}\ln(1+a_{n}) converges.
Homework Equations
The Attempt at a Solution
My first attempt is the direct...
Homework Statement
Check if the series below converge.
a) $$\sum_{n = 1}^\infty \frac{n}{2n^2 - 1}$$
b) $$\sum_{n = 2}^\infty (-1)^n \frac{2n}{n^2 - 1}$$
Homework Equations
The Attempt at a Solution
For a).
The series converge if the sum comes up to a finite value. If...
I'm looking for the proof of the following theorem/statement.
$$
\begin{align}
\lim_{n \to \infty} M_{X_n}(t) = M_X(t)
\end{align}
$$
for every fixed t \in \mathbb{R}.
My book only states the theorem without proving it and I haven't found a proof online. Any help is appreciated! :)
Assume the sequence of positive numbers ${a_n}$ converges to L. Prove that
$\lim_{n \to \infty} \sqrt[n]{a_1a_2...a_n} = L$ (The nth root of the product of the first n terms)
Since ${a_n}$ converges we know that for every $\epsilon> 0$ there is an $N$ such that for all $n > N$ $ |a_n -...
Define ##\rho(f)=\int |f|\mathrm d\mu## for all integrable ##f:X\to\mathbb C##. This ##\rho## is a seminorm, not a norm. Does ##\rho(f_n-f)\to 0## imply ##f_n\to f## a.e.?
I kind of think that it should, because in the case of real-valued functions, ##\int|f_n-f|\mathrm d\mu## is the area...
Homework Statement
Let f be a non-negative measurable function. Prove that
\lim _{n \rightarrow \infty} \int (f \wedge n) \rightarrow \int f.The Attempt at a Solution
I feel like I'm supposed to use the monotone convergence theorem.
I don't know if I'm on the right track but I created a...
Hi, I am writing to you to ask for help.
I have a problem with the contact "Frictional". I'll show you screenshots of my settings and program the console errors. Everything will be shown in the screenshots. The problem is that between the beams and the top element and the bottom element is...
Homework Statement
I know that the harmonic series is divergent.But why \frac{-1}{n} is also divergent?
I've search for some test to test that, but I could not find a method for negative series.
So how can I prove the series is divergent?:confused:
Homework Equations
The Attempt...
Homework Statement
Series:
\sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{(x)^n}{na^n}
what is the behaviour of the series at radius of convergence \rho_o=-z ?
Homework Equations
The Attempt at a Solution
So I can specify that the series is monatonic if z is non negative as...
Homework Statement
determine the radius of convergence of the series expansion of log(a + x) around x = 0
Homework Equations
The Attempt at a Solution
So after applying the Taylor series expansion about x=0 we get log(a) + SUM[(-1)^n x^n/(n a^n)] I understand how to get the...
Homework Statement
I've found that the typical way for using ratio test is to find the limit of an+1/an However, my tutor said that radius of convergence can be found by finding the limit of an/an+1 and the x term is excluded.
For example:Finding the interval of convergence of n!xn/nn
my...
I am currently learning series and testing for convergence. For comparison tests especially I am having an issue grasping the concept of picking a proper limit to compare too.
For example the following problem
If someone could please put it in the form where it actually looks like what it...
Homework Statement
Prove that the series \sum_{n=1}^{\infty}\frac{\sqrt{n+1}-\sqrt{n}}{n} converges.
The Attempt at a Solution
I think I'm going to use the comparison test but I'm having trouble coming up with a series to compare it to. Any clues would be great. Thanks!
a) Show that sum_(n=0)^infinity (2^n x^n)/((1+x^2)^n) converges for all x in R\{-1,1}
b) Even though this is not a power series show that sum above = 1 + sum_(n=1)^infinity (2nx^n) for all -1<x<=1.
For part a by the ratio and root test we get |(2x)/(1+x^2)| but this does not have an n in it...
Homework Statement
This is for Calculus II. We've just started the chapter on Infinite Series. n runs from 1 to ∞.
\Sigma\frac{1}{n(n+3)}
The Attempt at a Solution
I used partial fraction decomposition to rewrite the sum.
\frac{1}{n(n+3)}=\frac{A}{n}+\frac{B}{n+3}...
Determine whether the sequence converges or diverges, if it converges fidn the limit.
a_n = n \sin(1/n)
so Can I just do this:
n * \sin(1/n) is indeterminate form
so i can use lopitals
so:
1 * \cos(1/x) = 1 * 1 = 1
converges to 1?
Determine whether the sequence Converges or Diverges.
Tricky question, so check it out.
\frac{n^3}{n + 1}
So here is what I did
divided out n to get
\frac{n^2}{1} = \infty \therefore diverges
Now, here is what someone else did. They applied L'Hopitals, and then claimed that 3n^2 = \infty...
Determine whether the series is convergent or divergent.
\sum^{\infty}_{n = 1} \frac{n - 1}{3n - 1}
I ended up with \frac{1}{3} * 1 = \frac{1}{3} , which is 0.333 ... so wouldn't that mean that r < 1? Also wouldn't that mean that it is convergent since r < 1 ?
I don't understand why this is...