Convergence Definition and 1000 Threads

Homework Statement show a function f_n is not uniformly convergent using a theorem: Homework Equations if f_n converges uniformly to F on D and if each f_n is cont. on D, then F is cont. on D The Attempt at a Solution not really sure what to do. use the contrapositive? would that...

Homework Statement Hi there. Well, I was trying to determine the radius and interval of convergence for this power series: \displaystyle\sum_{0}^{\infty} \displaystyle\frac{x^n}{n-2} So this is what I did till now: \displaystyle\lim_{n \to{+}\infty}{\left...

Hi, I need to determine whether this improper integral converges or diverges \int_{-1}^{1} \frac{x}{\sqrt{1-x^2}}dx The original function DNE at -1, 1 so I split the limits \int_{-1}^{0} \frac{x}{\sqrt{1-x^2}}dx \ + \ \int_{0}^{1} \frac{x}{\sqrt{1-x^2}}dx I've...

I need to show that the by eliminating infinitely many terms of the harmonic series, the remaining subseries can be made to converge to any positive real numbers. I have no clue to prove this. I know harmonic series diverges really slowly, will this fact come into play? Thank you very much!

Homework Statement Prove\; that\;if\;\sum_{n=1}^{\infty} a_n \;converges,\;then \lim_{n\to\infty}a_n = 0 Book solution s_n= a_1 + a_2 +...+a_n s_{n-1}= a_1 + a_2 +...+a_{n-1} a_n=s_n-s_{n-1} Then they did a few limits, and proved that the difference is 0. BUt that is not my...

Homework Statement http://img840.imageshack.us/img840/3609/unleddn.png note that by log(n), i really mean NATURAL log of n Homework Equations it's convergent, but I can't figure out which test to useThe Attempt at a Solution there is no term to the nth power, so ratio test is useless; root...

Homework Statement f(x)=x^{0.4} Construct a power series to represent the function and determine the first few coefficients. Then determine the interval of convergence. The Attempt at a Solution Determining the first few coefficients is simple enough. Take the first few...

1. Let g_n (x)=nx*exp(-nx). Is the convergence uniform on [0, ∞)? On what subsets of [0,∞) is the convergence uniform? 3. I am looking for a proof of how the convergence is uniform (possibly using Weierstrass' M Test?). I understand that the subset that determines uniform convergence is...

I was reading Royden when I came across this cryptic statement:pg 222, "The concept of uniform convergence of a sequence of functions is a metric concept. The concept of pointwise convergence is not a metric concept." Can anyone illuminate this?

Homework Statement Is $\intop_{-\infty}^{\infty}\arctan(x)\, dx$ convergent? What about $\lim_{t\rightarrow\infty}\intop_{-t}^{t}\arctan(x)\, dx$?Homework Equations The Attempt at a Solution I think the first integral may actually be divergent the way its written and the second one...

Homework Statement How would I find the radius of convergence of this series? f(x)=10/(1-3x)2 is represented as a power series f(x)=\sum from n=0 to \infty CnXn Homework Equations The Attempt at a Solution Okay so I tried deriving, using d/dx(1/1-3x)=3/(1-3x)2 and ended up with...

Homework Statement Determine whether the series is absolutely convergent, conditionally convergent, or divergent. \sum (-1)^n\frac{e^{1/n}}{n^4} Homework Equations The Attempt at a Solution I used the root test so \sqrt[n]{\frac{e^{1/n}}{n^4}} --> \lim_{n\to \infty...

Homework Statement The summation from n=1 to infinity of ((n!)x^(2n))/((2n-1)!) Find the Interval of Convergence of this series. Homework Equations Ratio test The Attempt at a Solution I applied the ratio test, then got x^2 times the limit as n approaches infinity of...

Homework Statement This is the question of mine that I'm having a little confusion about. I know the whole process in which you use the ratio test to determine the radius of convergence and using that you test the end points of the summation to see if they converge at the end points aswell...

Homework Statement Show that if \vartheta is any constant not equal to 0 or a multiple of 2\pi, and if u_{0}, u_{1}, u_{2} is a series that converges monotonically to 0, then the series \sum u_{n} cos(n\vartheta +a) is also convergent, where a is an arbitrary constant. Homework...

Homework Statement Determine if the series the summation form n=2 to infinity of n/((n2+1)ln(n2+1)) is convergent or divergent. Homework Equations The Attempt at a Solution I applied the integral test and got positive infinity, so I it diverges. But I want to know if I'm right...

Homework Statement Find the radius and interval of convergence for the power series of n=0 of infinity of n^3(x-5)^n Homework Equations Ratio test: http://en.wikipedia.org/wiki/Ratio_test The Attempt at a Solution [(n+1)^3(x-5)^n+1 / n^3(x-5)^n] I am lost as to how to...

Homework Statement Does the series: sum from 1 to inf. (n+2)/(n+1) converge? If so does it converge absolutely? Homework Equations Ratio test for series The Attempt at a Solution I found this series to converge using the root test, (.jpg attached) however wolfram alpha...

Homework Statement Does ∫dx/sqrt(x^4+1) from x=-∞ to x=∞ converge or diverge? explain in detail if you can please. thanks Homework Equations limit comparison test direct comparison The Attempt at a Solution...well i have the answer, it converges. I just need a better...

Homework Statement Hi, So i don't know if this is a stupid question but i'll ask anyways. So I'm on the chapter where we start testing integrals for convergence. The books starts out with elementary functions then they move towards non elementary functions. Testing for them is OK, my problem...

EDIT: On pg. 390 of Kreyszig's Functional Analysis text, we have: "If T is a bounded linear operator on a nonempty Banach space, then the series \sum_{k=0}^\infty \left( \frac{T}{\lambda} \right)^k converges absolutely for |\lambda| > 2\| T \|." The argument presented in Kreyszig...

Homework Statement For which positive integers k is the following series convergent? (To enter - or , type -INFINITY or INFINITY.) Summation of n=1 to infinity of (n!)^2 / (kn)! Homework Equations ratio test: limit n-->infinity of [((n+1)!)^2/(kn+1)!] / [(n!)^2 / (kn)!] (have the...

Homework Statement I'm tasked with integrating the following functions, and values for t where the function converges: \int_{0}^{1}x^p\cdot ln(x) dxHomework Equations Integration by Parts Formula: \int udv=uv-\int vdu The Attempt at a Solution I found a definite integral...

Homework Statement A function f is defined by... f(x) = \frac{n+1}{3^{n+1}} x^n a.) find the interval of convergence of the given power series. b.) Find \lim_{x\rightarrow 0} \frac{f(x) - \frac{1}{3}}{x} c.) Write the first three nonzero terms and the general term for...

Homework Statement Determine the radius of convergence, the interval of convergence, and the sum of the series Summation from k=2 to ∞ of k(x-2)^k+1. Homework Equations ratio test? The Attempt at a Solution possibly take the derrivitive of the power series, then find the sum then integrate...

Homework Statement I need to find the domain of absolute convergence of the following series: ^{\infty}_{1}\sum(z+3)^{2n}/(2n)! Homework Equations Ratio test? The Attempt at a Solution I'm not really sure how to handle the complex variable z within the series. I attempted to use the...

Homework Statement http://im2.gulfup.com/2011-04-01/1301686351321.gif Homework Equations superlinearly convergence The Attempt at a Solution [PLAIN][PLAIN]http://im2.gulfup.com/2011-04-01/1301686616101.gif this is what i know about it, kindly help me

Homework Statement \Sigma (from index k = 1 until infinity) Within the Sigma is the series : (k! * (x^k)) Homework Equations Ratio Test : lim as k approaches infinity |a(k+1) / ak| The Attempt at a Solution When I apply the ration test to the series and simplify I get lim k...

Homework Statement Suppose that: sum [a_n (n-1)^n] is the Talyor series representation of tanh(z) at the point z = 1. What is the largest subset of the complex plane such that this series converges? Note: 'sum' represents the sum from n=0 to infinity Homework Equations tanh(z) =...

Homework Statement Discuss the convergence of Integral(x3/2 sin 2x dx) range is from 0 to 1Homework Equations|sin x| =< 1 The Attempt at a Solution The sine function converges absolutely. It is also increasing from 0 to 90 degrees, decreases until 27 decrease and it is negative from 180 to...

Homework Statement Say that \sum_{k=1}^{\infty }a_k converges and has positive terms. Does the following necessarily converge? \sum_{k=1}^{\infty }{a_k}^{5/4} Homework Equations If it necessarily converges, a proof is required, if not, a counter-example is required. The...

Homework Statement Let f_n be a sequence of holomorphic functions such that f_n converges to zero uniformly in the disc D1 = {z : |z| < 1}. Prove that f '_n converges to zero uniformly in D = {z : |z| < 1/2}.Homework Equations Cauchy inequalities (estimates from the Cauchy integral formula)The...

Find the convergent sum and find the sum of first five terms \sum_{n=1}^{\infty} \frac{sin(nx)}{2^nn} from 1 to infinity. I have found so far that: \sum_{n=1}^{\infty} \frac{sin(nx)}{2^n} = \frac{2sin(x)}{5-4cos(x)} I am not sure how to consider the \frac{1}{n} term. Can someone please help?

Homework Statement Determine if the following converge: a. (∞,n=1) ∑ (1+1/n)^n b. (∞,n=1) ∑ sin(n)/(n^2 + √n) c. (∞,n=3) ∑ 1/(k ln^2 k) The Attempt at a Solution a. I tried the root test, but it failed, so i immediately went to the limit divergence test...ended up getting...

Homework Statement Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. \sumn=1infinity (-3)n-1/4nHomework Equations A geometric series, \sumn=1infinity arn-1=a + ar + ar2 + ... is convergent if |r|< 1 and its sum is \sumn=1infinity arn-1 =...

Homework Statement Determine whether the sequence converges or diverges. If it converges, find the limit. an = e1/n Homework Equations The limit laws, adapted for sequences. The Attempt at a Solution I have the solution; I was just wondering if someone might explain it to me. I...

Homework Statement Does the series ((-1)^n*n!)/(1*6*11*...*(5n+1)) from n = 0 to \infty absolutely converge, converge conditionally or diverge? Homework Equations The Attempt at a Solution I did the ratio test for ((-1)^n *n!)/(5n+1)) and I found that it diverges but apparently...

Homework Statement \Sigma from n=0 to infinity (-10)n/n! Determine the absolute convergence, convergence, or divergence of the series. Homework Equations In this section, it's suggested that we use the following to determine a solution: A series is called absolutely convergent if the series...

Homework Statement \sum from n=1 to inf (1+ 1/2 + ... 1/n)x^n Find the radius of convergence and the interval of convergence of the given power series. Homework Equations Dunno.. The Attempt at a Solution Stuck thinking about it. I'm not sure if I can combine what's in brackets with the...

Homework Statement For what values of r does \int(from 0 to infinity) xre-x dx converge? I assume that the problem refers to r as any real number. 2. The attempt at a solution I have given this a try but I am really not confident that I did it right... First i used integration...

Homework Statement If the sequence xn ->a , and the sequence yn -> b , then xn - yn -> a - b The Attempt at a Solution Can someone check this proof? I'm aware you cannot subtract inequalities, but I tried to get around that where I indicated with the ** in the following proof...

Homework Statement Test for convergence \sum sin(1/n) / \sqrt{ln(n)} sum from 2 to inf Homework Equations Limit comparison test if lim an / bn to inf doesn't equal 1 you know if it converges or diverges by limit comparison test The Attempt at a Solution I've tried a lot of different...

Homework Statement show if has a uniform convergence of pointwise also we know that x gets values from 0 to 1The Attempt at a Solution for the pointwise I think its easy to show that limfn(x) as n->infinity is 0 but I am really stuck in uniform convergence I know that fn converges...

My functional analysis professor made the following assertion the other day: If f_n \to f in the L^2 norm, then there is a subsequence f_{n_k} that converges pointwise almost everywhere to f. This is the first I've heard of that...can someone point me to a proof of this proposition? Does it have...

Homework Statement Homework Equations [PLAIN]http://img577.imageshack.us/img577/6756/physforumsquestion.png The Attempt at a Solution I am pretty much stuck guys. Any help will be greatly appreciated!

Homework Statement Let r be a positive number and define F = {u in R^n | ||u|| <= r}. Use the Componentwise Convergence Criterion to prove F is closed.Homework Equations The Componentwise Convergence Criterion states: If {uk} in F converges to c, then pi(uk) converges to pi(c). That is, the...

Hello, I am preparing for a screening exam and I'm trying to figure out some old problems that I have been given. Given: Suppose f is contained in L1([a,b]) Prove for almost everywhere x is contained in [a,b] limit as h goes to 0+, int (abs(f(x+t)+f(x-t)-2f(x)))dt = 0 Initially I...

Homework Statement Approximate the sum of the series S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n!} by calculating S_10. Estimate the level of error involved in this problem. AND S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n^4} Approximate the sum of the series by...

Homework Statement Show that there are continuous functions g:[-1,1]\to R such that no sequence of polynomials Q_n satisfies Q_n(x^2)\to g(x) uniformly on [-1,1] as n\to\infty The Attempt at a Solution Suppose there is a sequence Q_n such that Q_n(x^2)\to g(x) uniformly for g(x)=x. Then...

Homework Statement [PLAIN]http://img153.imageshack.us/img153/4822/radiusm.jpg Homework Equations The Attempt at a Solution Using the ratio test: \left | \frac{e^{i(n+1)^2 \theta} \theta^{n+1} z^{(n+1)^2}}{e^{in^2 \theta} \theta ^n z^{n^2}} \right | = | \theta...