Convergence Definition and 1000 Threads

Homework Statement Let Ʃanx^n and Ʃbnx^n be two power series and let A and B be their converging radii. define dn=max(lanl,lcnl) and consider the series Ʃdnx^n. Show that the convergence radius of this series D, is D=min(A,B) Homework Equations My idea is to use that the series...

Homework Statement Does the series \Big( \sum_{n=1}^\infty\frac{1}{(3^n)*(sqrtn)} \Big) Converge or Diverge? By what test?Homework Equations 1/n^p If p<1 or p=1, the series diverges. If p>1, the series converges. If bn > an and bn converges, then an also converges. The Attempt at a...

Hello everyone, I need some help on doing convergence tests (comparisons I believe) on some Ʃ sums. I have three, they are: 1. Ʃ [ln(n)/n^2] from n=1 to ∞. I tried the integral test but was solved to be invalid (that is, cannot divide by infinity). Therefore I believe it to be a...

|f(x0)f''(x0)|<|f'(x0)|^2 where I is the interval containing the approximate root x0, is the convergence criterion of ... (a) Newton - Raphson method (b) Iteration method (c) Secant method (d) False position method According to me its (a), but I confused because this formula is not directly...

This thread will be dedicated to discuss the convergence of various definite integrals and infinite series , if you have any question to post , please don't hesitate , I hope someone make the thread sticky. 1- $$\int^{\infty}_0 \left(\frac{e^{-x}}{x}...

Here is the question: Here is a link to the question: How to find Radius of Convergence for Sum of ((x-3)^n)/(n3^n) from n =1 to inf? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

$$\int^{\infty}_{0} \frac{\log(t) \sin(t) }{t} \, dt$$ Can we say the following : $$\int^{\infty}_{0} \frac{\log(t) \sin(t) }{t} \, dt=\int^{\epsilon}_{0} \frac{\log(t) \sin(t) }{t} \, dt+\int^{\infty}_{\epsilon} \frac{\log(t) \sin(t) }{t} \, dt$$ 1-$$\int^{\epsilon}_{0} \frac{\log(t) \sin(t)...

Homework Statement My main issue here is that if I use the ratio test I end up with lim (x(n!+1))/((n+1)!+1) n-> ∞ I don't know how to progress here. I believe that the limit will equal 0 and so it's interval of convergence is from -∞<x<∞ with a Radius of convergence of ∞. Is it safe for me...

Alright, I have my final for Calc 2 on Monday. I am only stuck on two sections of problems because I am terrible at convergence and divergence/alternating series. I have questions for each one really. They are as follows: 4) a. Can I simply factor out the alternating series and apply a...

Homework Statement Ʃ(((-1)^n)(x^n))/(n+1) from 0 to ∞Homework Equations The Attempt at a Solution I took applied the ratio test and got that lim|(x^(n+1))/(n+2) * (n+1)/(x^n)| =|x| so that means for it to converge |x|<1 Radius of convergence is 1 My interval is (-1<x<1) Now I check the...

How do you show that $\displaystyle \int_{0}^{\infty} \Big( \frac{x^{a-1}}{\sinh x} - x^{a-2} \Big) \ dx $ converges for $0 \le a <1$?

Homework Statement Let g(x)= (2/3)*(x+1/(x^2)) and consider the sequence defined by pn= g(pn-1) where n≥1 a) Determine the values of p0 \in [1,2] for which the sequence {pn} from 0 to infinity converges. b) For the cases where {pn} converges (if any), what is the rate of convergence...

Homework Statement The coefficients of the power series \sum_{n=0}^{∞}a_{n}(x-2)^{n} satisfy a_{0} = 5 and a_{n} = (\frac{2n+1}{3n-1})a_{n-1} for all n ≥ 1 . The radius of convergence of the series is: (a) 0 (b) \frac{2}{3} (c) \frac{3}{2} (d) 2 (e) infinite Homework EquationsThe Attempt at...

Homework Statement Let (a_n) be a sequence. (i) Prove that if \sum\limits_{n = 1}^\infty {{a_n}} converges, then \sum\limits_{n = 1}^\infty {\left( {{a_{2n - 1}} + {a_{2n}}} \right)} also converges. (ii) Prove that if \sum\limits_{n = 1}^\infty {\left( {{a_{2n - 1}} + {a_{2n}}} \right)}...

Here is the question: Here is a link to the question: Determine the interval of convergence? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

1. Determine a power series, centered at zero for the function ∫f(x)dx. Identify the interval of convergence. f(x) = ln(x+1) = ∫\frac{1}{x+1} 2. 3. i found the power series, which is : Ʃ ((-1)^(n))(x^(n+1)) / (n+1) Im okay with that, but i need help on finding the interval of...

Test these for convergence. 5. infinity E...((n!)^2((2n)!)^2)/((n^2 + 2n)!(n + 1)!) n = 0 6. infinity E...(1 - e ^ -((n^2 + 3n))/n)/(n^2) n = 3 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?

Test these for convergence. 3. infinity E...((-1)^n)*(n^3 + 3n)/((n^2) + 7n) n = 2 4. infinity E...ln(n^3)/n^2 n = 2 note: for #3: -((n^2 + 3n))/n) is all to the power of e Btw, E means sum. Which tests should I use to solve these?

Test these for convergence. 1. infinity E...n!/(n! + 3^n) n = 0 2. infinity E...(n - (1/n))^-n n = 1 Btw, E means sum. Which tests should I use to solve these?

Homework Statement The question is: Suppose that lim x_k=x_*, where x_* is a local minimizer of the nonlinear function f. Assume that \triangledown^2 f(x_*) is symmetric positive definite. Prove that the sequence \left \{ f(x_k)-f(x_*) \right \} converges linearly if and only if \left...

Here is the question: Here is a link to the question: Calculus Power Series/Radius of Convergence/Interval of Convergence Question? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement The nth partial sum of the series ∞ Ʃ an n=1 is given Sn = 5-1/n Determine weather the series is convergent or divergent The Attempt at a Solution Looked in my book on how to do this one. couldn't find anything on it. What i was thinking...

Homework Statement limit as k--> infinity ABS VALUE((cos(pi*k+pi))/(cos(k*pi))) Homework Equations The Attempt at a Solution Can someone prove to me why this limit is equal to 1? I have tried several other sources and I have not had any luck.

Give an example of a sequence \{ f_n\} of continuous functions defined on [0,1] such that \{ f_n\} converges pointwise to the zero function on [0,1], but the sequence \{ \int^{1}_{0} f_n\} is unbounded. I'm pretty lost on this one.

Homework Statement Find the interval of convergence for (x-10)^{n}/10^{n} The Attempt at a Solution If i use the ratio test on this, i end up with (x-10)/10, which doesn't make sense to me since there is no "n" in this result. Is there another method i need to be using?

I have constructed mono layer slab of rutile TiO2 (110) with 11 Angstroms vacuum between two layers. I used the Quantum wise software for the construction of the slab. Then I have exported the Quantum espresso input from Quantum wise software. Now the problem is the SCF calculations are not...

Homework Statement Determine either absolute convergence, conditional convergence or divergence for the series.Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{(-1)^n}{5n^{1/4} + 5}The Attempt at a Solution It converges conditionally i know, but i can't figure out how. 1. I applied the...

Homework Statement Absolute, Conditional, - convergence, or Divergence.Homework Equations \displaystyle \sum^{∞}_{n=1} (-1)^n e^{-n} The Attempt at a Solution 1. Alternating Series Test 2. Ratio Test for ABsolute Convergence 1. \displaystyle (-1)^n (1/e)^n an > 0 for n=1,2,3,4 - YES...

Homework Statement Find the interval of convergence.Homework Equations \displaystyle \sum^{∞}_{n=0} \frac{(x-5)^n}{n^4 * 2^n}The Attempt at a Solution I used the ratio test as follows: \displaystyle \frac{(x-5)^{n+1}}{(n+1)^4 * 2^{n+1}} * \frac{n^4 2^n}{(x-5)^n} taking the limit: (x-5) lim...

Homework Statement Determine the radius of convergence and the interval of convergence og the folling power series: n=0 to infinity Ʃ=\frac{(2x-3)^{n}}{ln(2n+3)} Homework Equations Ratio Test The Attempt at a Solution Well I started with the ratio test but I have no clue where...

I am trying to determine convergence for the series n=1 to infinity for cos(n)*pi / (n^2/3) and I am doing the Ratio Test. I found the limit approaches 1 but is less than 1. Does this mean that the limit = 1 or is < 1? I am somewhat confused since this changes it from inconclusive to convergent.

Homework Statement http://gyazo.com/55eaace8994d246974ef750ebeb36069 Homework Equations Theorem III : http://gyazo.com/af2dfeb33d3382430d39f275268c15b1 The Attempt at a Solution At first this question had me jumping to a wrong conclusion. Upon closer inspection I see the...

I have a question on which test to use for series n=1 to infinity for (-1)^n / (n^3)-ln(n) in order to determine convergence/divergence. I am pretty sure I determined it converges through the Alternating Series Test(correct me if I'm wrong) but I am not sure whether it is conditional or...

Hello. I have a problem. I don't know how to finish my proof. Maybe someone of you will be able to help me. So task is: Suppose that fn and gn are function of the sequence. fn simply convergence(? don't know if english term is same) to f, and gn simply convergence to g. Need to prove that...

Homework Statement Suppose that ##s_n(x)## converges uniformly to ##s(x)## on ##[b, ∞)##. If ##lim_{x→∞} s_n(x) = a_n## for each n and ##lim_{n→∞} a_n = a## prove that : ##lim_{x→∞} s(x) = a## Homework Equations ##\space ε/N## The Attempt at a Solution I see a quick way to do this one...

I'm a bit confused about how my book defines convergence. Definition: A sequence {an} convergences to l if for every ε > 0 there is a natural number N such that, for all natural numbers n, if n > N, then l a,-l l < ε note, l a,-l l = the absolute value Maybe someone could give me an...

If ##p## is a limit point of ##E## then ##\exists \ (p_n) \ s.t. (p_n) \rightarrow p## For the sequence construction, can I just define ##(p_n)## as such: ##For \ q \in E, \ \ define \ (p_n) := \left\{\Large{\frac{d(p,q)}{n}} \right\}_{n=1}^\infty##

I'm not sure if this should go in the homework section, since I'm essentially asking for textbook style problems. Anyways, if you know of any good integrals/series convergence problems off the top of your head, could you give some to me? I guess if I can't figure any out I'll post them as...

Construct extrapolation table with optimal rates of convergence Homework Statement Let S be a cubic spline interpolant that approximates a function f on the given nodes x_{0},x_{1},...,x_{n} with the boundary conditions: S''(x_{0})=0 and S'(x_{n})=f'(x_{n}). Use S to estimate f(0.1234567)...

Homework Statement Converge or diverge?Homework Equations \displaystyle a_{n} = \frac{(-1)^{n+1}}{2n-1}The Attempt at a Solution all i know is to perhaps take ln of numerator and denominator to get the exponent down below?

Homework Statement Let ##s_n(x) = \frac{1}{n} e^{-(nx)^2}##. Show there is a function ##s(x)## such that ##s_n(x) → s(x)## uniformly on ##ℝ## and that ##s_n'(x) → s'(x)## for every x, but that the convergence of the derivatives is not uniform in any interval which contains the origin...

Homework Statement Determine the convergence, both pointwise and uniform on [0,1] for the following sequences : (i) ##s_n(x) = n^2x^2(1 - cos(\frac{1}{nx})), x≠0; s_n(0) = 0## (ii) ##s_n(x) = \frac{nx}{x+n}## (iii) ##s_n(x) = nsin(\frac{x}{n})## Homework Equations ##s_n(x) →...

Homework Statement Test for convergence or divergence. Homework Equations \displaystyle\int_0^∏ {\frac{sin∅}{\sqrt{pi-∅}} d∅} The Attempt at a Solution the solution manual does the following: pi -∅ = x \displaystyle\int_0^pi {\frac{sinx}{\sqrt{x}} d∅} 0 <=...

Homework Statement Let {a_n} be a sequence | (a_n+1)^2 < (a_n)^2, 0 < (a_n+1) + (a_n). Show that the sequence is convergent Homework Equations n/a The Attempt at a Solution So I am feeling like monotone convergence theorem is the way to go there. It seems to me that (a_n+1)^2 <...

Homework Statement I would like to use the Weierstrass M-test to show that this family of functions/kernels is uniformly convergent for a seminar I must give tomorrow. H_{t} (x) = \sum ^{-\infty}_{\infty} e^{-4 \pi ^{2} n^{2} t} e^{2 \pi i n x} . Homework Equations The Attempt at a...

Simple question: Are there any series which we don't know whether or not they converge?

Homework Statement Consider the function f_n(x)=n\cdot I\left[|x|<\frac{1}{2n}\right] considered as a distribution in D'(\mathbb{R}), where I denotes the indicator function. Recall that f_n converges to \delta_0, the delta distribution, in D'(\mathbb{R}). Show that f_n^2-n\delta_0...

Homework Statement Really tough series to work with. Determine the convergence ( absolute or conditional ) or divergence of : ##\sum_{n=2}^{∞} \frac{(-1)^n n}{n^p + (-1)^n}## Homework Equations ?? Series tests? The Attempt at a Solution This series is really ugly. I'm not sure how to...

Hi everyone, Im running the KF to learn parameters of a model, the log likelihood of the p(Y_{k}|Y_{k-1}), however decreases. Can anyone advise, does this mean my implementation is wrong or can this just be the case. Advice appreciated Thanks

A while back, I dug out a topic I worked on many years ago after taking a course in ordinary differential equations and I was left with an unanswered question, which I thought I would post here. While my question arose from studying numeric methods for approximating the solutions to ODEs, I feel...