Convergent Definition and 334 Threads

If a sequence of operators \{T_n\} converges in the norm operator topology then: $$\forall \epsilon>0$$ $$\exists N_1 : \forall n>N_1$$ $$\implies \parallel T - T_n \parallel \le \epsilon$$ If the sequence converges in the strong operator topology then: $$\forall \psi \in H$$...

Homework Statement I'm trying to find out whether or not this sequence diverges or converges. If it converges, then what's the limit. {4+sin(1/2*pi*n)} The Attempt at a Solution This one is a bit confusing to me since sin oscillates between 1 and -1. So if you plug in (pi*infinity)/2, that...

Homework Statement Show, from the definition of what it means for a function to converge to a limit, that the sequence ##\left\{x^t\right\}_{t=1}^{\infty}## with ##x^t = \frac{2t+5}{t^2+7}## converges to ##0## as ##t## goes to infinity. Homework Equations A sequence converges to ##x^0 \in X##...

Mod note: Moved from a homework section. 1. Homework Statement this is my lecturer's notes, he says it is a divergent series, but this seems like an obvious convergent series to me.. could someone verify? Homework Equations https://www.dropbox.com/s/mc5rth0cgm94reg/incorrect maths.png?dl=0...

Homework Statement Show that ##\sum \frac {cos(\frac{n\pi} {3})} {n^2}## is absolutely convergent, and therefore convergent Homework Equations Comparison test to 1/n^2 The Attempt at a Solution So to be absulutely convergent the absolute value of the series needs to be convergent. So we...

Homework Statement Determine whether the sequence converges or diverges. If it converges, find the limit. Here's the sequence: http://www4a.wolframalpha.com/Calculate/MSP/MSP89541ea2ag9dg617bcd6000050d52e94i67ei593?MSPStoreType=image/gif&s=39&w=66.&h=44. Homework Equations N/A The Attempt at...

I was goofing around with Mathematica and found that Sum_(k>=1)(sin(k)/k)=Sum_(k>=1)(sin(k)/k)^2. In other words a convergent series such that if you square each of its terms the sum is the same. Question is: is this a unique property or are there other convergent series with the property? Cheers.

Homework Statement For which of them will the corresponding fixed point iteration xk+1 = g(xk) be locally convergent to the solution xbar in [0, 1]? (The condition to check is whether |g'(xbar)| < 1.) A) 1/x2 -1 B)... C)... compute xbar to within absolute error 10-4. Homework Equations 3. The...

Hi, I am working on investigating an idea I proposed regarding a ramjet that operates in subsonic flow (of a fixed speed) with a convergent intake. That utilizes the pressure immediately behind a standing shock-wave for compression. I have posted a link to my initial report here and I now need...

Homework Statement Wolfram alpha and integral calculator say zero, but my school's HW on webworks does say it's divergent. I don't suppose it's wrong either, since the issue is probably a technicality Arguably, on desmos.com the graph is pretty damn near odd. Homework Equations Int(-inf)(inf)...

Hi, I hope that, given I've sourced what I can, you may be able to help? I'm Currently working on a lab report for my Aeroengines unit based on Convergent-Divergent nozzle (http://imgur.com/HQML2Au). From the drawing, I have been provided D1, D2, D3, D4, T1, T2, T3, T4, P1, P2, P3, Velocity...

I'm not sure if this is true or not. but from what I can gather, If the set of Natural numbers (divergent sequence) {1, 2, 3, 4, 5,...} is broken up to say {1}, is this a subsequence that converges and therefore this statement is true?

hi everyone ! i have a question , Can a numerical convergent series be a convergent functional series? I know only the other way that a convergent functional series can be a convergent numerical series because i take a function as a constant so i have a numerical series and it converges.

Homework Statement Is the sequence {(n!)/(n^n)} convergent or divergent. If it is convergent, find its limit. Homework Equations Usually with sequences, you just take the limit and if the limit isn't infinity, it converges... That doesn't really work here. I know I'm supposed to write out the...

Homework Statement I want to prove that if X is a normed space, the following statements are equivalent. (a) Every Cauchy sequence in X is convergent. (b) Every absolutely convergent series in X is convergent. I'm having difficulties with the implication (b) ⇒ (a). Homework Equations Only...

Hi, Mathematica is telling me the value of this series, but I can't figure out how to do it on paper. Can someone please explain? $$\sum_{n=0}^{\infty}\frac{1}{(2n+1)^2}=\frac{\pi^2}{8}$$

Homework Statement The problem and solution are attached as TheProblemAndSolution.jpg. Homework Equations Definition of the limit of a sequence. The Attempt at a Solution I understand how P = ϵ + |A| can be seen as an upper bound that proves that the sequence is bounded, but for the last bit...

Homework Statement 1+\dfrac{2^2}{2!}+\dfrac{3^2}{3!}... \infty The Attempt at a Solution t_n = \dfrac{n^2}{n!} \\ \dfrac{n}{(n-1)(n-2)...1} I tried applying the Ratio Test but couldn't find another function which would give me a finite limit when divided by that function.

Determine whether the integral is convergent or divergent. Evaluate those that are convergent. $\int_{0}^{9} \ \frac{1}{\sqrt[3]{x-1}},dx$ $\int_{-\infty}^{\infty} \ \cos\left({\pi t}\right),dt$ how do i determine whether it's conver/diver?

Homework Statement Which of the series, diverge or converge ∑ 5^n/(4^n +3 ) Homework Equations The Attempt at a Solution Taking the limit as n→∞ we have (5^n ln 5)/ (4^n ln 4) , my question is here how does it become like this, which part am I missing here?

Suppose there is a sequence xn=1/(n-2). We know we n tend to infinity the sequence tends to zero. But at n=2 it is equal to infinity. Is this sequence convergent? There is also a theorem that all convergent sequence are bounded for every n. But the sequence above is not bounded at n=2...

determine if series is absolutely convergent, conditionally convergent, or divergent \sum^{\infty}_{n = 1} (n^2 + 9)(-2)^{1-n} which i turned into \sum^{\infty}_{n = 1} (n^2 + 9)(-2)^{-n+1} so using the ratio test I got: \frac{((n+1)^2 + 9)(-2)^{-n})}{n^2 + 9 * (-2)^{1-n}} which ended...

Determine if the positive term series is convergent or divergent \sum^{\infty}_{n = 1} \frac{n + cosn}{n^3 + 1} can't I just ignore the cosn and look at it like this: \sum^{\infty}_{n = 1} (-1)^n \frac{n}{n^3 + 1} Then can't I just look at it as n--> \infty and see that I end up with...

Homework Statement Is the following convergent or divergent? ln(2(n+1))-ln(2n) Homework Equations comparison test The Attempt at a Solution I put it into the form ln(1+(1/n)), but I don't understand what to use as Bn for the comparison test. (which is what wolfram alpha uses)

Hello PF. Homework Statement A function f is defined by f(x) = 1 + 2x + x2 + 2x3 + x4 + 2x5 + x6... Find the radius of convergence of the series and the explicit formula for f(x).Homework Equations The Attempt at a Solution I know that the formula for the series is going to be similar to the...

Homework Statement Problem is attached in this post. Homework Equations Problem is attached in this post. The Attempt at a Solution I know how to find the sum of this series, but I don't know what method to use in order to prove that this series converges. Also I don't understand how I'm...

Homework Statement an= (n/n+2)^n ANS: 1/e^2 The Attempt at a Solution I was told this was convergent and I need to find the limit of the sequence. How do I do this, as I seem to keep getting that this is divergent. Isn't it divergent to infinity? Or am I missing something?

Homework Statement an= (n/n+2)^n The Attempt at a Solution I was told this was convergent and I need to find the limit of the sequence. How do I do this, as I seem to keep getting that this is divergent. Isn't it divergent to infinity? Or am I missing something?

How can I tell if the following series is Divergent or Convergent: ∑( e^(6pi*n) sin^2(4pi*n) ) the sum limits are from -infinity to infinity

Homework Statement Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. ∫ from negative infinity to infinity of (x^8*e^-x^9) The Attempt at a Solution The answer is diverged to infinity. But I got that by guessing. Can someone explain to me why...

Homework Statement Is there a real number c such that the series: ∑ (e - (1+ 1/n)^n + c/n), where the series goes from n=1 to n=∞, is convergent? The Attempt at a Solution I used the ratio test by separating each term of the function as usual to find a radius of convergence, but that doesn't...

A limit of a sequence is definitely convergent if: If for any value of K there is an N sufficiently large that an > K for n > N, OR for any value of K there is an N sufficiently large that an<±K for n > N My only question is what exactly are K, N, an and n? What values are they? How would...

Homework Statement Consider the sequence ak = (3)/(4^(2k)). Show is convergent, find sum. Please check work. Homework Equations ak = 3/(4^2k) let s {n} be the series associated with the sequence. Cannot write summation notation here, but k starts at 1 (k = 1 on bottom) and infinity...

Homework Statement Please look over my work and tell me if I did something wrong. Suppose Bn is a divergent sequence with the limit +∞, and c is a constant. Prove: lim cBn -> ∞ = +∞ for c > 0 Homework Equations N/A The Attempt at a Solution lim Bn -> ∞ = means that for some value K >...

I am currently designing a convergent nozzle for use in experiments and wanted to check something: Will the pressure and density of the flow always expand to ambient (following the isentropic relations) when it reaches the nozzle exit regardless of nozzle contraction ratio and length? Any...

Homework Statement 1. Determine if arctan(7+1/n)-arctan(7) converges or diverges 2. Determine if 2/1-1/2-1/3+2/4-1/5/-1/6+2/7... converge or diverge Homework Equations series tests The Attempt at a Solution 1.My gut instinct is to do limit comparison test w/ 1/n, and it...

Homework Statement As a part of Method of Frobenius, I am encountered with the following problems: Evaluate the following limits: Q1. \stackrel{limit}{_{x→0}}\frac{1-2x}{x} Q2. \stackrel{limit}{_{x→0}}\frac{x-1}{x} Q3. \stackrel{limit}{_{x→0}}\frac{1-2x}{x}+\frac{x-1}{x} In context of the...

Homework Statement Ʃ ne(-n2) Homework Equations The Attempt at a Solution I used the ratio test and wanted to know if the way I did it is correct or not |a(n+1) / a(n)| n+1 (e(-n2 -2n-1)) / n e(-n2) Now e-n^2 cancels and we get limn→∞ n+1/n * 1/(e2n)(e) After you take the limits you get...

Homework Statement . Let ##\{a_n\}_{n \in \mathbb N}## a sequence of real numbers such that ##lim_{n \to \infty} a_n=0## and let ##b_n=a_n+2a_{n+1}-a_{n+2}##. Prove that ##\sum_{n=1}^{\infty} a_n## is convergent iff ##\sum_{n=1}^{\infty} b_n## is convergent. The attempt at a solution...

Homework Statement Solve the following : a) Show that ordp((p^n)!)=1+p+p^2+p^3+...+p^(n-1) b)For which values of p does the following series converge in Qp? 1)1+(15/7)+(15/7)^2+(15/7)^3+... 2)1!+2!+3!+4!+... 2. The attempt at a solution For a) I want to to count how...

was looking at a proof of this here: http://gyazo.com/8e35dc1a651cec5948db1ab14df491f8 I have two questions, why do you set K = max of all the terms of the sequence plus the 1 + |A| term? Why do you need the absolute value of all the terms? i.e. why |a_1| instead of |a_1|?

Homework Statement 2.11. Determine (explicitly) a convergent subsequence of the sequence in R2 given for n = 1; 2; : : : by xn =(e^{n}sin(n\pi/7),((4n+3/3n+4)cos(n\pi/3)) I know that the Bolzano-weierstrass theorem says that every bounded sequence has a convergent subsequence. I...

I quote a question from Yahoo! Answers I have given a link to the topic there so the OP can see my response.

In my book it says that the series (1+x)^n converges for x<1. However I put n = -1 and wolfram says that the series does not converge. However if I let x = 1/y where y>1 then the expansion of (1+1/y)^-1 is equal to: (which I will define as (SERIES 1)) 1 - y + (1/y)2 - (1/y)3 + (1/y)4...

Homework Statement The question : http://gyazo.com/7eb4b86c61150e4af092b9f8afeaf169 Homework Equations Sup/Inf axioms Methods of constructing sequences ##ε-N## ##lim(a_n) ≤ sup_n a_n## from question 5 right before it. I'll split the question into two parts. The Attempt at a...

Is it posible that the product of two different divergent series be a convergent serie?

Homework Statement Ʃ √n/(ln(n))^2 from n=2 to ∞ Homework Equations Series Test for convergent and divergent The Attempt at a Solution I tried doing ratio test and gotten [√(n+1)*(ln(n))^n] / [(ln(n+1))^(n+1) * √n] to find the limit, do...

1. Find the sum of the convergent series: ∞ Ʃ 2/[(4n-3)(4n+1)] n=12. Hm... Okay, so I started with the nth term test, and the denominator gets huge very fast. So I'm pretty sure it goes to zero. So that tells us nothing other than that it does not FOR SURE diverge. Since it has no n in the...

Hello, Imagine a convergent nozzle; static pressure at exit is atmospheric. The fluid is air. The pressure on the pressurized side is P. goal #1: achieve nozzle exit velocity somewhat below sonic goal #2: have P as high as possible. Is this possible to achieve through the geometry of...

Hello. I.m struggling to understand how to test for convergent and divergent. ∞ Ʃ (n/(n+1))^(n^2) n=1