Convergence Definition and 1000 Threads

Homework Statement Does the following sequence converge, or diverge? a_{n} = sin(2πn) Homework Equations The Attempt at a Solution \lim_{n→∞} sin(2πn) does not exist, therefore the sequence should diverge? But it actually converges to 0? I appreciate all help thanks. BiP

consider the rational function : f(x,z)=\frac{z}{x^{z}-1} x\in \mathbb{R}^{+} z\in \mathbb{C} We wish to find an expansion in z that is valid for all x and z. a Bernoulli-type expansion is only valid for : \left | z\ln x \right |<2\pi Therefore, we consider an expansion around z=1 of the form...

Hello, Could anyone please assist in proving that given a sequence converges it has a maximum/minimum/both? I have hitherto written that granted it converges it must be bounded and have a supremum and an infimum. Now, how may I proceed to prove that the latter are indeed within (the...

Homework Statement Be K \geq 1. Conclude out of the statement that \lim_{n \to \infty } \sqrt[n]{n} = 1, dass \sqrt[n]{K} = 1 The Attempt at a Solution \lim_{n \to \infty } \sqrt[n]{K} \Rightarrow 1 \leq \sqrt[n]{K} \geq 1 + ... I got issues with the right inequality...

Homework Statement Please write a specific function to define this series. Also provide a sum that the series converges to.Homework Equations Sn - {1, 1+1/e2, 1+1/e2+1/e4, 1+1/e2+1/e4+1/e6, ...} The Attempt at a Solution I know that the common ratio is 1/e2 and that you can raise that to...

Homework Statement Let x_n be a convergent sequence with a ≤ x_n for every n, where a is any number. Prove that a ≤ lim x_n when n→∞. Homework Equations Definition of limit. The usual ε, N stuff. The Attempt at a Solution Let lim x_n = x and choose ε=x_n-a. Hence we have |x_n -...

Homework Statement Let A be a finite subset of R. For each n in N, let x_n be in A. Show that if the sequence x_n is convergent then it must become a constant sequence after a while. Homework Equations The definition of limit. The Attempt at a Solution As A is finite, at least...

Homework Statement Use the ratio test to find the radius of convergence and the interval of convergence of the power series: [[Shown in attachment]] Homework Equations an+1/an=k Radius of convergence = 1/k Interval of convergence: | x-a |∠ R The Attempt at a Solution I...

Homework Statement Show that \int^{\infty}_{-\infty} \frac{e^{kx}}{1+e^{x}}dx converges if 0<k<1 Homework Equations None The Attempt at a Solution Well if I can show that the integral is dominated by another that converges then I'm done, but I haven't been able to come up with one...

Homework Statement Study the convergence of the following sequences a_{n} = \int^{1}_{0} \frac{x^{n}}{1+x^{2}} b_{n} = \int^{B}_{A} sin(nx)f(x) dx The Attempt at a Solution For the first one, I said it was convergent. I'm not exactly sure why though, my reasoning was...

OK... "A power series can be differentiated or integrated term by term over any interval lying entirely within the interval of convergence" When i do term by term differentiaion or t-by-t integration of a series though, am i making use of this fact? Does this come into play later in a...

Hello everyone! I am told that the limit of $\frac{x_{n+1}}{x_n}$ is $L>1$. I am asked to show that $\{x_n\}$ is not bounded and hence not convergent. This is what I got so far: Fix $\epsilon > 0$, $\exists n_0 \in N$ s.t. $\forall n > n_0$, we have $|\frac{x_{n+1}}{x_n}-L|<\epsilon$...

1. Is 1/(√(2n-1) convergent? 2. I have tried the first comparison test: an= 1/(√(2n-1) and bn=1/(n1/2. 0<=an<=bn. But bn diverges so we get no information. I have tried the second comparson test and let bn=1/n. But an/bn=∞ so once again I get no information. I have tried the ratio...

Reading through a real analysis textbook I noticed that \sum 1/K diverges but \sum 1/K1+\epsilon converges for all \epsilon > 0. This is confusing because 1/k will eventually be equally as small as the terms in 1/K1+\epsilon and therefore it should also converge. It may take much longer but...

Homework Statement x_{n+1} = (x_{n} + 2)/(x_{n}+3), x_{0}= 3/4Homework Equations The Attempt at a Solution I've worked out a few of the numbers and got 3/4, 11/15, 41/56, 153/209, ... It seems to be monotone and bounded below indicating it does converge I think. I need help figuring out what...

What is the statement of the mean square convergence of Fourier series?

Prove the Schwarz's and the triangle inequalities for infinite sequences: If $$ \sum_{n = -\infty}^{\infty}|a_n|^2 < \infty\quad\text{and}\quad \sum_{n = -\infty}^{\infty}|b_n|^2 < \infty $$ then $\sum\limits_{n = -\infty}^{\infty} a_nb_n$ converges absolutely. To show this, wouldn't I need to...

Here's the deal... I don't understand the limit as n→∞ of [(1+(.05/n))^20n -1]/[.05/n] My Calculus book says that it's supposed to approach {e^[(.05)(20)]-1}/[.05], but the numerator is a constant while the denominator goes to 0 as n→∞. The textbook, by Dr. Gilbert Strang, has similar limits...

I'm wondering about uniform convergence. We're looking at it in my complex analysis class. We are using uniform convergence of a series of functions, to say that we can interchange integration of the sum, that is: \int\sum b_{j}z^{j}dz=\sum\int b_{j}z^{j}dz=\int f(z)dz On an intuitive level I...

Homework Statement I need to show that f_{n}=sin(\frac{z}{n}) converges uniformly to 0. Homework Equations So I need to find K(\epsilon) such that \foralln \geq K |sin(\frac{z}{n})|<\epsilon I'm trying to prove this in an annulus: \alpha\leq |z| \leq\beta The Attempt at a Solution I'm having...

[b]1. Homework Statement [/b I want to see if a sequence converges by deciding on monotonicity and boundness. The sequence is: an=(n+1)/(2n+1) How to I go about determining if it converges or not based on those two factors? I am lost on how to go about it. THanks for any help...

Homework Statement \sum(\frac{2n}{2n+1})n2 (The sum being from n=1 to ∞). Homework Equations The Attempt at a Solution Used exponent properties to get (\frac{2n}{2n+1})2n. Using the root test, the nth root of an = lim n->∞(\frac{2n}{2n+1})2 = 1. However, the root test is...

Suppose that nx is binomially distributed: B((n-1)p, (n-1)p(1-p)) I wish to find the expected value of a function f(x), thus \sum_{nx=0}^{n-1} B() f(x) Assume that f() is non-linear, decreasing and continuous, f(x) = c is [0,1] to [0, ∞) I want to show that the above sum converges to f(p)...

Homework Statement If sequence an diverges to infinity and sequence an*bn converges then how do I prove that sequence bn must converge to zero? Homework Equations The Attempt at a Solution I really don't know how to go about this so any help would be so appreciated. Thanks

Hi all, I wonder if the following are equivalent. 1) E(|X|) < infinity 2) I(|X| > n) goes to 0 as n goes to infinity (I is the indicator function) 3) P(|X| > n) goes to 0 as n goes to infinity. 1) => 2) and 2) => 3) are easy to see, please help me to show 2) => 1) and 3) => 2)...

Let f(x) be a continuous functions on [0,∞) and that ∫ |f(t)|^2dt is convergent for 0≤t<∞. Let ∫ |f(t)|^2dt for 0≤t<∞ equals F. Show that lim(σ→∞) ∫(1-x/σ)|f(x)|^2 dx for0≤x≤σ converges to F. I know that it needs to prove that lim(σ→∞) ∫(x/σ)|f(x)|^2 dx for0≤x≤σ converges to 0. Can anyone...

I am trying to prove a theorem related to the convergence of Fourier series. I will post my proof below, so first check it and then my question will make sense. Is there any flaw in my proof? Also, here I proved it for integrable functions monotonic on an interval on the left of 0. But what if...

The sum of "(n+3)!/(3n+2)!" with n=1 to n=inf. How do I find if it converges or diverges by using one of the tests(ratio, roots series, divergence, etc)?

Where is the fallacy in this "proof" that the Fourier series of f(x) converges to f(x) if f is continuous at x and has period 2π? (I read in Wikipedia that a counterexample had been provided). Start with the Dirichlet integral for the N-th partial sum of the (trigonometric) Fourier series...

Given a power series \sum a_n x^n with radius of convergence R, it seems that the series converges uniformly on any compact set contained in the disc of radius R. This might be a silly question, but what's an example of a power series that doesn't actually also converge uniformly on the whole...

Homework Statement Prove that the sequence {a_n} converges to A if and only if lim n--->∞ (a_n-A)=0. Homework Equations The Attempt at a Solution It's an if and only if proof, but I'm not sure how to prove it. Please help!

Homework Statement Test the following integral for convergence \int^{2}_{0}\frac{dx}{1-x^{2}} Homework Equations The Attempt at a Solution So far I have brought it down to \int^{2}_{0}\frac{1}{1-x}+\frac{1}{1+x} dx However, it seems that this integral produces a...

Homework Statement Test the following integral for convergence: \int^{∞}_{-∞}\frac{dx}{\sqrt{x^{4}+1}} Homework Equations The Attempt at a Solution I was able to use the ratio test to show that the integral converges if and only if \int^{∞}_{-∞}\frac{dx}{x^{2}} converges, but I haven't...

Hi, I am trying to self study analysis and was practicing some problems. I wasn't sure if this solution to one of the problems I came across was rigorous enough. Basically, by writing down the first few terms of 3^n and n!, I figured I can say 3^n < 3*(n-1)! for all n>=13...without...

Homework Statement Consider sequences {an} and {bn}, where sequence {bn} converges to 0. Is it true that sequence {anbn} converges to 0? The Attempt at a Solution Proof. First I assumed (an) is bounded, and so there exists M > 0 such that |an| < M for all n 2 {1, 2, 3, . . .}...

Homework Statement If xn-> ∞ then xn/xn+1 converges. Homework Equations The Attempt at a Solution I can see why the statement is true intuitively, but do not know how to make a rigorous proof. I have looked at the definitions of divergence/convergence but can get any ideas of...

So I know how to find the "Interval of Convergence" for a power series representation of a Function f(x). But I Still don't know what that "Interval of Convergence" does for me other than I can choose a number in it and plug it into the series. For Example[SIZE="4"] e^{x}=\sum^{∞}_{n=0}...

NOT talking about nonabsolute vs absolute convergence. I'm talking about conditional convergence. In my analysis text, this was a bit that was covered as enrichment and it straight up blew my mind. I don't get it. How can you simply rearrange terms and come up with a separate sum? They showed a...

Homework Statement xn+1 = xn + cosxn , n>=1 where x0 E [π/4 , 3π/4] = D. Show it converges, find rate of convergence.Homework Equations contraction theoremThe Attempt at a Solution Setting a function f(x) = x+cosx we have f'(x) = 1 - sinx, f''(x)= -cosx. Now f' >= 0, so f is increasing. For...

1. Find the radius and the interval of convergence for the series: Ʃ n=2 --> inf : [(-1)nxn]/ [4nln(n)] 2.To find the radius, we use the alternating series test. **an+1/an 3. From the alternating series test I find that the limit as n --> inf = 4. So our radius is 4...

Pretend that you are expaining the following to someone who knows nothing about complex numbers and within a universe where complex numbers have not been invented. In examining the function f(x) = \frac{1}{1 + x^2} we can derive the series expansion \sum_{n=0}^\infty (-1)^n x^{2n} We...

Hi, All: If {f_n}:ℝ→ℝ are measurable and f_n-->f pointwise, then convergence is a.e. uniform. Are there any conditions we can add to have f_n-->f in some open interval (a,b)? Correction: convergence happens in some subset of finite measure; otherwise above not true.

Prb:(x^4+4*x^2+16)y"+4(x-1)y'+6xy=0 P=(x^4+4*x^2+16) Q=4(x-1) R=6x P=0 for - 1 - 3^(1/2)*i 1 - 3^(1/2)*i - 1 + 3^(1/2)*i 1 + 3^(1/2)*i Q=0 for 1 R=0 for 0 Do we ignore Q & R, plotting P, then find shortest distance which would equal 2?

Homework Statement show that the integral of the poisson kernel (1-r^2)/(1-2rcos(x)+r^2) converges to 0 uniformly in x as r tend to 1 from the left ,on any closed subinterval of [-pi,pi] obtained by deleting a middle open interval (-a,a) Homework Equations the integral of poisson...

Hi all, http://www.scribd.com/doc/100079521/Document-1 Actually, I am trying to learn monotone convergence theorem, and I am stuck at one specific point, on the first page it says that ∫-∞→∞ f_n(x)dx = 1 for every n but the almost everywhere limit function is identically zero, what does it...

Hello all, Again I find myself at odds with my online class. Somehow, and with two problems in a row, I am finding the reciprocal answer to what Math Lab is telling me. I would be very appreciative is someone could check my work. Find the limit of convergence, and the radius. \sum...

Hello, I'm having hard times with the following simple linear ODE coming from a control problem: $$u(t)' \leq \alpha(t) - u(t)\,,\quad u(0) = u_0 > 0$$ with a given smooth α(t) satisfying $$0 \leq \alpha(t) \leq u(t) \quad\mbox{for all } t\geq 0.$$ My intuition is that $$\lim_{t\to\infty}...

Homework Statement The quotient test can be used to determine whether a series is converging or not. The full description is in the attachment. Homework Equations The Attempt at a Solution ( i ) Why must they both follow the same behaviour? Even if p ≠ 0, it says nothing about...

Homework Statement Does this series converge absolutely or conditionally?Homework Equations Series from n=1 to ∞ (-1)^(n+1) * n!/2^n The Attempt at a Solution In trying to apply the alternating series test, I have found the following: 1.) n!/2^n > 0 for n>0 2.) Next, in testing to see if...

Hi, while reading some artificial intelligence book, i came upon the following sum. How can I evaluate it analytically, so not guess it by computing many terms? It's easy to see by ratio test that it converges (intuitively too, since its a linear vs exponential function). \sum_{i=1}^\infty...