Convergence Definition and 1000 Threads

Homework Statement [PLAIN]http://img225.imageshack.us/img225/7501/complexh.jpg Homework Equations The Attempt at a Solution How do I show this absolutely converges?

Find the value of x so that the series below converge.[/b] \sum 1/ [(k^x) * (2^k)] (k=1 to \infty) Using ratio test, I 've got [(1/2) * 1^x] < 1 for all x in R But when I use different value of x, series converge and diverge! Really need your help! Thank you so much

I know that the sequences meets the following: (n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n) I've got the feeling that this sequence is alternating or decreasing, but I was unable to prove it. Usually I use induction to prove things about such inductive sequence but in this case I don't have real values t o...

Hello, I have question about using Dirichlet's Convergence Test which states: 1. if f(x) is monotonic decreasing and \lim_{x\rightarrow \infty} f(x)=0 2. G(x)=\int_a^x g(t)dt is bounded. Then \int_a^\infty f(x)g(x)dx is convergent. But what about the following situation: f(x)=1/x g(x)=cosxsinx...

Homework Statement Ok, so I don't need help with this part, I just got stuck at the following step when attempting to find the interval of convergence: The Attempt at a Solution I got here: -4 < x^2 < 4 So, I need to solve this inequality. But can I? How can I take the square root...

Hi. Can I have some help in answering the following questions? Thank you. Let {f_n} be a sequence of functions from N(set of natural numbers) to R(real nos.) where f_n (s)=1/n if 1<=s<=n f_n (s)=0 if s>n. Define f:N to R by f(s)=0 for every s>=1...

It's been a while since I've done rate of convergence problems, how would i find the rate of convergence for either of these sequences? 1) limn->infsin(1/n)=0 2)limn->infsin(1/n^2)=0

it's a new semester, and we're at it again. series abounds! Homework Statement Let . Determine whether the following series converges: Homework Equations -1 \leq \sigmak \leq 1 The Attempt at a Solution i feel like the series inherently diverges because of the 1/k element...

Homework Statement Find the radius of convergence of the power series: a) \sum z^{n!} n=0 to infinity b) \sum (n+2^{n})z^{n} n=0 to infinity Homework Equations Radius = 1/(limsup n=>infinity |cn|^1/n) The Attempt at a Solution a) Is cn in this case just 1? And plugging it in...

Hi. I have been banging my head against this problem for a while and I just don't get it. Maybe (probably) it's something wrong with my logarithm-fu or limit-fu. I just registered to ask this because I couldn't find an answer anywhere, and I've been reading these forums for a while for other...

Hi and sorry if I misplaced the thread. I'm having quite some trouble with analyzing the convergence of the following series : Homework Statement :[/B] Determine whether the series is convergent or divergent, absolutely & normally. \sum (-1)^n * [e-(1+1/n)^n] Homework Equations...

Hi and sorry if I misplaced the thread. I'm having quite some trouble with analyzing the convergence of the following series : \sum (-1)^n * [e-(1+1/n)^n] I had troubles both with absolute and normal convergence. With normal convergence I tried Leibniz 1) lim a(n) = 0 Which is ok { lim...

I am not very familiar with terms from numerical analysis, thus I do understand the definition for convergence rate from http://en.wikipedia.org/wiki/Rate_of_convergence" . Still, here the definition appears only for sequences. Which is the definition for rate of convergence for functions? For...

Homework Statement I need a power series with a radius = pi. (So when you do the ratio test on this power series you get pi) Homework Equations The Attempt at a Solution I tried x^n*sin(n) and thought of stuff like that but couldn't come up with a working power series

Suppose I have two sequences of r.v.s Xn and Yn. Xn converges to X with probability 1, and Yn converges to Y with probability 1. Does (Xn, Yn) converges to (X, Y) with probability 1? Is there a reference to confirm or negate this? Thanks a lot.

Homework Statement If \suman2 and \sumbn2 converge show that \sumanbn is absolutely convergent Homework Equations The Attempt at a Solution I think I should do something with the statement 2ab\leq a^2 + b^2

Homework Statement (\sqrt{(n+1)} - \sqrt{n} ) / \sqrt{n} I'm trying to show this series converges.Homework Equations divergence test: as n approaches infinity, if the sequence does not approach 0 then the series diverges ratio test: as n approaches infinity, if the ratio between subsequent...

"Convergence Factors" In all my textbooks I always see these random convergence factors thrown in (+0's or +i*nu or some such) but I have never seen a book that would dirty itself by steeping so low as to explain what they are (I'm looking at you Wen, Bruus and Flensberg, Fetter and Walecka...

Hi, well let me put the question a bit clear...my concern area is on ode and pde...my question is when you solve a pde/ode analytically and get a solution by asymptotic means does this mean that if solution exists then ...when using convergence as an alternative way of getting solution of the...

hello, does convergence imply asymptotic relation of an ordinary differential equation?

I was reading a book about Calculus that I came to a problem that the author claimed convergence of a series won't change if we subtract a finite number of its terms from it, It seems to be intuitively clear, but I need a proof. so please Prove that the convergence/divergence status of a series...

Homework Statement This is a nice one, if it's correct. Show that if (fn) is a sequence of elements of C(X, Y) (where Y is a metric space) which converges uniformly, then the collection {fn} is equicontinuous. The Attempt at a Solution Let ε > 0 be given and let x0 be a point of X...

Homework Statement Hi All, I've been having great difficulty making progress on this problem. Suppose gn converges to g a.e. on [0,1]. And, for all n, gn and h are integrable over [0,1]. And |gn|\leqh for all n. Define Gn(x)=\intgn(x) from 0 to x. Define G(x)=\intg(x) from 0 to x...

Homework Statement Show that if x=(x1, x2,...) and y=(y1, y2,...) \in \ell2 then The sum from i=1 to infinity of |xiyi| converges Homework Equations The Attempt at a Solution since x and y are elements of l^2 then The sum from i=1 to infinity of (xi)2 and (yi)2 both converge...

Homework Statement Could anyone help me with Laurent series? I do not understand it at all even though the book has several examples. And here is one with my comments Find the Laurent series of \frac{1}{(z-1)(z-2)} a in the region abs(z) < 1 b in the region 1< abs(z) < 2 c in the...

Is it possible for a series to converge without the constraint that a_n+1< or equal to a_n? Can we have a convergent series with only the requirement a_n >0 and the limit as x approaches infinity = 0 (i.e. not a decreasing monotonic series)? If yes include 3 series which disprove the...

Hello everyone. I was hoping someone could clarify this "heuristic" argument I found online. First, what is the analytic function they speak of and is its derivative difficult to compute? Second, does this look like a legit argument? : If you take the derivative w.r.t s of both sides of sum...

Hi, I need to know how one can check space and time convergence using Richardson Extrapolation. Does anyone know any good references. I have a slight idea... the thing I am wondering about is how using this method can eliminate the need for further simulations using smaller time steps or a...

Please teach me this: Can we demontrate the convergence of perturbation series of quantum field theory(Feymann diagrams) after making the renormalizing procedure? If we can't demontrate that,why we still consider the perturbative method using in quantum field theory being useful and believable...

1. Homework Statement So i have to solve this integral with dominate convergence theorem. How can i prove that the sequence f_{n} it s monotone? \lim_{n \rightarrow +infty} \int_{0}^{+infty} \frac{1 -sin(\frac{x}{n})}{\sqrt(x^2 +\frac{1}{2}}

I have found an interesting infinite sum that appears to converge to a number. In fact, I have created four of these using slightly different rules. Since, all of the odd primes are involved in each series, I know that the series are infinite. I will confirm this by using Euler's Prime...

Hi all, Due to my idiotic uncapability with mathematical software, I'm unable in using Mathematica (8.0) for a very trivial issue. I've a set from different countries data and I have to study a descriptive convergence, but I'm unable. The matter is simply putting the dates on a plot and work...

Homework Statement If we find the convergence rate of a numerical method, does this rate have to be positive? and are there any conditions eg rate must be less than 1? Homework Equations The Attempt at a Solution Thank you

Homework Statement Well, hi there. I have to study the convergence of the next series using the comparison criteria, or the comparison criteria through the limit of the quotient. \displaystyle\sum_{n=1}^\infty{\displaystyle\frac{3n^2+5n}{2^n(n^2+1)}} I think that I should use the...

Homework Statement v=sum of i*(i+1)/((1+y)^i) from i =1 to i=infinity for y > 0 Homework Equations The Attempt at a Solution This isn't a homework question per se (and I'm not that interested in the actual number solution), but how would one go about solving v? Are there any...

Homework Statement Find the series' radius and interval of convergence. What what value of x does the series converge absolutely, conditionally? Sum (n=0 to infinity) (nx^n)/((4^n)((n^2) +1))) Homework Equations The Attempt at a Solution Not quite sure where to start with this...

If you follow QG research you will remember that in September 2009 there was a mainly Loop QG school/workshop on Corfu at which several people presented minicourses (John Baez, Carlo Rovelli, Abhay Ashtekar, Vincent Rivasseau, John Barrett...) Each minicourses was a series of 5 onehour lectures...

does anyone know where i can find a decent bank of convergence and taylor series problems and solutions? my calc book is a bit lacking, and i don't have the solutions manual.

on page 48 of baby Rudin, it says " the sequence {1/n} converges in R1 to 0, but fails to converge in the set of all positive real numbers [with d(x,y) = |x-y|]." ok, I know it has something to do with 1/n going to infinity near zero, but it does that whether the metric space is R1 or just...

Homework Statement I need help to decide if the series below are convergent or divergent. \sum_{k=2}^{\infy}(\frac{1}{ln(k!)}) Homework Equations The Attempt at a Solution I tried using the d'Alembert ratio test but the ratio is 1 if I calculated it correctly and then nothing can be said...

Generally when I post, it's with a specific problem. However in this case, the issue I'm running into is that I don't even have the slightest idea of how to even setup the problem to attempt it. Honestly, I can't seem to find this anywhere in my book and I'm not sure what's really being asked...

Homework Statement Prove if {bn} converges to B and B ≠ 0 and bn ≠ 0 for all n, then there is M>0 such that |bn|≥M for for all n. Homework Equations What I have so far: I know that if {bn} converges to B and B ≠ 0 then their is a positive real number M and a positive integer N such...

Homework Statement Prove the series converges uniformly on R to functions which are continuous on R. \sumn\geq0 (-x)2n+1/(2n+1)! Homework Equations The Attempt at a Solution I'm having trouble actually figuring out what to use for this series.. It looks like a Taylor series...

Homework Statement Use the comparison test to explain why the series (3^n)/(2^n + 4^n) is convergent. He said specifically not to use the limit comparison test on this one. Homework Equations 1/2^n The Attempt at a Solution I know I should be comparing it to 1/2^n because it is a...

Hi, Im really stuck on my homework . The question is : For what values of x do you expect the following Taylor series to converge? Do not work out the series . (a) sqrtX^2-x-2 about x=1/3 b) sin(1-x^2) about x=0 for a) I've put no vlues of x would the series converge. is this correct? and...

Homework Statement Let x1 > 9000, and xn+1 = )2009xn + 2010)/2011 for n >1 show that (xn) converges and find its limit Homework Equations Definition of a limit, Monotone Convergence Theorem. The Attempt at a Solution Since xn+1 is monotone for n>1 and bounded, then it...

Homework Statement The problem asks you if the series: 1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}... converges or diverges Homework Equations The Attempt at a Solution I tried to apply the Leibniz rule but I realized it can't be applied. Is there a transformation of this...

Homework Statement let f be a continuous function on an interval around 0, and let an=f(1/n) (for large enough n). prove that if f''(0) exists and f'(0)=f(0)=0, then converges Homework Equations i proved earlier in the problem that if the series converges, f(0)=0, and if f'(0) exists...

Homework Statement I have stared at this too long and do not know which test to approach it with, even writing it out. The problem is State the Convergence or Divergence of the given series: Summation n=1 to Infinity of 1 / sqrt (n^3 + 2n) Homework Equations I narrowed it down to...

Homework Statement let f be a continuous function on an interval around 0, and let an=f(1/n) (for large enough n) prove that if f'(0) exists and converges, then f'(0)=0 Homework Equations proved earlier in the problem that if the series converges, then f(0)=0. The Attempt at a...