Convergence Definition and 1000 Threads

Homework Statement Ʃn!(x-1)n I need to find the radius of convergence for this summation from n=0 to n=∞ The Attempt at a Solution I started off with the ratio test: (n!(n+1)(x-1)(x-1)n)/(n!(x-1)n) = (n+1)(x-1) (x-1)lim(n+1)...Now at this point it looks to me like the series does...

[answered] I want to know why this particular approach is wrong so I can learn from my mistakes. Homework Statement a_n = \frac{ln(n^3)}{2n}The Attempt at a Solution For the sake of being time efficient, I will skip writing things like the limit as n approaches infinity etc. a_n =...

I know that for any a>0 and k,t\in\mathbb{R}, the integral \int_0^a t^k\; dt converges if and only if k>-1. Is it true that if k is complex then \displaystyle \int_0^a |t^k| \; dt converges if and only if \text{Re}(k)>-1 since if t is real, |t^k| does not depend on the imaginary part of k?

Determine the values of the complex parameters p and q for which the beta function \int_0^1 t^{p-1} (1-t)^{q-1}dt converges absolutely. The solution says: Split the integral into 2 parts: \displaystyle \int_0^1 t^{p-1} (1-t)^{q-1}dt = \int_0^{1/2} t^{p-1} (1-t)^{q-1}dt + \int_{1/2}^1...

Suppose h(x) is a continuous function for x > 0. If \int^∞_1{h(x)dx} converges then for constant 0 < a < 1, \int^∞_1{h(\frac{x}{a})dx} also converges. The answer is true. Anyone care to explain why? I would have chosen false, because I was thinking that h(x/a) is larger than h(x) so we...

I have some problems here with Series and Convergence... Here are the problems and my guesses at it. http://img822.imageshack.us/img822/9523/23341530.png It won't tell me which one is wrong, but it just says one/all is wrong. Any help is appreciated. Attempts at solving, I tried...

Homework Statement I am struggling to answer this question please help Find the region of convergence for the following complex series and draw the region Ʃ(i+z)^(2n-1)/2^(2n+1)The Attempt at a Solution Here is my hand written working, sorry i could figure out how to use the symbols...

Homework Statement Let f be a continuous function on [1,∞) such that \lim_{x\rightarrow ∞}f(x)=α. Show that if the integral \int^{∞}_{1} f(x)dx converges, then α must be 0. Homework Equations Definition of an Improper Integral Let f be a continuous function on an interval [a,∞). then we...

Suppose there exists a sequence f_n of square-integrable functions on \mathbb R such that f_n(x) \to f(x) in the L^2-norm with x \ f_n(x) \to g(x), also in the L^2-norm. We know from basic measure theory that there's a subsequence f_{n_k} with f_{n_k}(x) \to f(x) for a.e. x. But my professor...

Homework Statement Suppose that the power series \sumanxn for n=0 to n=∞ has a radius of convergence R\in(0,∞). Find the radii of convergence of the series \sumanxn2 from n=0 to n=∞ and \sumanx2n.Homework Equations Radius of convergence theorem: R = 1/limsup|an|1/n is the radius of...

Hello all. I have to present a proof to our Intro to Topology class and I just wanted to make sure I did it right (before I look like a fool up there). Proposition Let c be in ℝ such that c≠0. Prove that if {an} converges to a in the standard topology, denoted by τs, then {can}...

Has anybody got any idea as to how to prove that Ʃ 1/(n(log(n))^p) converges? (where p>1)

Suppose I want to determine the convergence of ((sin(n))^4)/(1+n^2) using limit comparison test. I divide it by 1/(1+n^2). All that remains is (sin(n))^4. Now as the limit goes to infinty, the range of values (sin(n))^4 can give is 0 to 1. Now it gives many more values above zero then at zero...

Homework Statement Suppose that the following series converges when x = -4 and diverges when x = 6. ∑{n=0 -> ∞} c_n • x^n What is the interval of convergence? The Attempt at a Solution I think it is [-5,5) but my friend reckons that it is [-5,6). I don't think [-5,6) is correct because this...

I am wondering if this is solvable. Determine the convergene/divergence of the sum from n=1 to infinity of sec(n)/n. All the tests appear to fail and listing out the sequence of partial sums produces no useful results.

i see people discussing the convergence radius of a perturbation series in the literature i am really baffled generally, one can only get the first few coefficients of a perturbation series that is, the perturbation series is not known at all how can one determine the convergence...

NEVERMIND! IT IS 0! I SOMEHOW WAS STARING AT THE WRONG ANSWER SHEET FOR A LITTLE BIT! THANK YOU! 1. Homework Statement Determinte whether the sequence converges or diverges: (n^2)/(e^n)2. Homework Equations The book says that the solution is: e/(e-1). However, the limit of the equation...

Homework Statement Hello. I am trying to prove a result that I have been making use of, but never really proved. Consider the recurrence equation x(k+1) = 0.5 x(k) + u(k), where u(k) is a bounded sequence. For this problem, assume that u(k) goes to zero. I want to prove that x(k) goes to...

Homework Statement f(x)= {1, ‐1/2<x≤1/2} {0, ‐1<x≤ ‐1/2 or 1/2<x≤1} State whether or not the function's Fourier sine and cosine series(for the corresponding half interval) converges uniformly on the entire real line ‐∞<x<∞ Homework Equations The Attempt at a Solution...

Homework Statement So what I was taught was that if the lim of the ratio test is the series is always absolutely convergent. If it is >1 the series is always divergent. But if it is =1 then we don't know. So would that mean that all conditionally convergent series would have a limit = 1? I...

The radius of convergence of \sum\limits_{k=1}^\infty\displaystyle\frac{z^n}{n} is 1. It converges on all of the boundary \partial B(0,1) except at z=1. One way of looking at this is to analyse \sum\limits_{k=1}^\infty\displaystyle\frac{\cos n\theta}{n}+\frac{\sin n\theta}{n}. You can see the...

Let $f(x)=-x$ for $-l\le x\le l$ and $f(l)=l.$ a) Study the pointwise convergence of the Fourier series for $f.$ b) Compute the series $\displaystyle\sum_{n=0}^\infty \frac{(-1)^n}{(2n+1)}.$ c) Does the Fourier series of $f$ converge uniformly on $\mathbb R$ ? ------------- First I need to...

Homework Statement Here is the problem: http://dl.dropbox.com/u/64325990/HW%20Pictures/integraltest.PNG The Attempt at a Solution I know it is convergent because it is very similar to 1/n^1.5 which is convergent as well. However what would I compare this with using the Integral Test to...

Hey guys, I can't get his question dealing with orders of convergence at all so any help would be nice. Q: Find the conditions on A so that the iteration $$x_{n+1}=x_n-Af(x_n)$$ will converge to a root of f if stared near the root. I know I should look at the taylor series expansion of f about...

Homework Statement \sum from n=1 to n=\infty (1 + \frac{x}{n})n2 Determine the values of x for which the series converges absolutely, converges conditionally and diverges.The Attempt at a Solution So i tried using the root test for the absolute value of (1 + \frac{x}{n})n2, but it was...

Homework Statement f_{n} is is a sequence of functions in R, x\in [0,1] is f_{n} uniformly convergent? f = nx/1+n^{2}x^{2} Homework Equations uniform convergence \Leftrightarrow |f_{n}(x) - f(x)| < \epsilon \forall n>= n_{o} \inN The Attempt at a Solution lim f_{n} = lim...

Homework Statement Show that a sequence ##f_n \to f \in C[0,1]## with the sup norm ##|| ||_\infty##, then ##f_n \to f \in C[0,1]## with the integral norm. The Attempt at a Solution given ##\epsilon > 0 \exists n_0 \in N## s.t ##||(fn-f) (x)|| < \epsilon \forall n > n_0## with ##...

Homework Statement Let Y_i be standard normal random variables, and let X be an N vector of random variables, X=(X_1, ..., X_N) where X_i = 1/{sqrt{N}} * Y_i. I want to show that as N goes to infinity, the vector X becomes "close" to the unit sphere. Homework Equations The...

Homework Statement k is a positive integer. \sum^{\infty}_{n=0} \frac{(n!)^{k+2}*x^{n}}{((k+2)n)!} Homework Equations The Attempt at a Solution I have no idea.. this is too confusing. I tried the ratio test (which is the only way I know how to deal with factorials) but I get...

Homework Statement The series: \sum^{n=\infty}_{n=0} \frac{(-1)^{n+2}(x^{3}+8)^{n+1}}{n+1} Homework Equations The Attempt at a Solution using the ratio test, I get the following: |x^{3}+8|<1, but I know that the radius of convergence must be in the form: |x-a|<b, where...

Homework Statement here is the series: \sum^{\infty}_{n=0}x(-15(x^{2}))^{n} Homework Equations The Attempt at a Solution I know that -1<-15x^{2}<1 for convergence (because of geometric series properties) but I run into a problem here: -1/15<x^{2}<1/15 You can't...

Hi, I'm new to posting questions on forums, so I apologise if the problem is poorly described. My problem is solving a simulation of the state of a mineral processing froth flotation plant. In the form x@i+1 = f(x@i), f represents the flotation plant. f is a computationally intensive solution...

Homework Statement Use integration, the direct comparison test, or the limit comparison test to test the integrals for convergence. If more than one method applies, use whatever method you prefer. Homework Equations ∫sinθdθ/√π-) The Attempt at a Solution I don't know which method...

Hi, Suppose I have an analytic function f(z)=\sum_{n=0}^{\infty} a_n z^n the series of which I know converges in at least |z|<R_1, and I have another function g(z) which is analytically continuous with f(z) in |z|<R_2 with R_2>R_1 and the nearest singular point of g(z) is on the circle...

Homework Statement Let f,g be continuous on a closed bounded interval [a,b] with |g(x)| > 0 for all x in [a,b]. Suppose that f_n \to f and g_n \to g uniformly on [a,b]. Prove that \frac{1}{g_n} is defined for large n and \frac{f_n}{g_n} \to \frac{f}{g} uniformly on [a,b]. Show that this is...

Folks, I am looking at this task. 1) What does it mean to say a sequence converges in a normed linear space? 2) Show that if a sequence fn converges to f in C[0,1] with sup norm then it also converges with the integral norm? Any idea on how I tackle these? thanks

Homework Statement The original question is posted on my online-assignment. It asks the following: Determine whether the following series converges or diverges: \sum^{\infty}_{n=1}\frac{3^{n}}{3+7^{n}} There are 3 entry fields for this question. One right next to the series above...

Homework Statement Assume a_n is a bounded sequence with the property that every convergent sub sequence of a_n converges to the same limit a. Show that a_n must converge to a. The Attempt at a Solution Could I do a proof by contradiction. And assume that a_n does not converge...

Homework Statement Q) Summation from 1 to infinity (1+(-1)^i) / (8i+2^i) This series apparently converges and I can't figure out why. Homework Equations The Attempt at a Solution (1+(-1)^i) / i(8+2^i/i) Taking the absolute value of the above generalization...

Homework Statement how to prove that radius of convergence of a sum of two series is greater or equal to the minimum of their individual radii i don't know how to begin, can someone give me some ideas?

How do I determine whether the following sequences of complex functions converge uniformly? i) z/n ii)1/nz iii)nz^2/(z+3in) where n is natural number

Homework Statement Let I=[a,b], f : I to R be continuous and suppose that f(x) >= 0 . If M = sup{f(x):x ε I} show that the sequence $$\left( \int_a^b (f(x))^n \, dx \right)^\frac{1}{n}$$ converges to M The Attempt at a Solution Where do I start? I'm thinking of having g_n(x)=...

Do the following series converge or diverge? a) $\displaystyle \sum\frac{1}{n!}\left(\frac{n}{e}\right)^n$ b) $\displaystyle\sum\frac{(-1)^n}{n!}\left(\frac{n}{e}\right)^n$ My attempt: a) $\displaystyle\frac{1}{n!}\left(\frac{n}{e}\right)^n\approx\frac{1}{\sqrt{2\pi n}}$ (Stirling’s formula)...

Homework Statement Does pointwise convergence imply convergence of integral on C[0,1]? Homework Equations The Attempt at a Solution Would like to verify that f(t)=t^n is a counter example of this. Pointwise convergence goes to 0 on [0,1) and 1 on [1]. Norm(1) is unbounded. I...

Using the fact that the interval $\displaystyle\left[2k\pi+\frac{\pi}{4},\ 2k\pi+\frac{3\pi}{4}\right]$ contains an integer, prove that $\displaystyle\sum\frac{\sin n}{n}$ is not absolutely convergent.

Homework Statement Find the radius of convergence and interval of convergence of the series: as n=1 to infinity: (n(x-4)^n) / (n^3 + 1) Homework Equations convergence tests The Attempt at a Solution i tried the ratio test but i ended up getting x had to be less than 25/4 ...

Homework Statement Show that \sqrt{2},\sqrt{2\sqrt{2}},\sqrt{2\sqrt{2\sqrt{2}}} converges and find the limit. The Attempt at a Solution I can write it also like this correct 2^{\frac{1}{2}},2^{\frac{1}{2}}2^{\frac{1}{4}},2^{\frac{1}{2}}2^{\frac{1}{4}}2^{\frac{1}{8}} so each time i...

Prove that if $\displaystyle \left|\frac{a_{n+1}}{a_n}\right|\le\left|\frac{b_{n+1}}{b_n}\right|$ for $\displaystyle n\gg 1$, and $\displaystyle\sum b_n$ is absolutely convergent, then $\displaystyle\sum a_n$ is absolutely convergent.

Does the following series converge? $\displaystyle\sum\frac{n^5}{2^n}$

Test for convergence: $\sum_2^\infty\frac{1}{n(\ln n)^p}$ when $p<0$. My working: Consider the function $f(x)=\frac{1}{x(\ln x)^p}=\frac{(\ln x)^q}{x}$ when $q=-p>0$. In order to use the integral test, I have to establish that $f(x)$ is decreasing for $x\ge 2$. How do I proceed?