Convergence Definition and 1000 Threads

Homework Statement Find the radius of convergence of sum from 1 to n of 1/(n^n) * x^(2^n) Homework Equations The Attempt at a Solution Clearly ratio test isn't going to work straight away. I'm not sure how to deal with the 2^n exponent

Use a comparison test to determine whether the series \sum (n+1)/(n^{2}+n+1) diverges or converges. I started out by simplifying the series to 1/n+1 and then from there I compared it to 1/n, which converges. 1/n is greater than 1/n+1 so based on the comparison test, the original series...

Homework Statement I'm this series to see if it's convergent or divergent. I tried the root test, but it came out inclusive, and now I am trying to figure out if the ration test works. The only thing I'm asking which would be the right test for this. Homework Equations ∞Ʃn=1...

Homework Statement Given that limit of s_{2n} is L and limit of s_{2n+1} is L, prove that lim s_{n} is also L. Homework EquationsThe Attempt at a Solution This seems very obvious: If the even terms of a sequence approach a number and the odd terms of that sequence approach the same number...

Homework Statement I have a problem set that asks me to determine, first, the radius of convergence of a complex series (using the limit of the coefficients), and second, whether or not the series converges anywhere on the radius of convergence. Homework Equations As an example: Σ(z+3)k2 with...

Hello everyone, I have some problems with the convergence for the planewave cutoff energy, I tried to find the correct PWcE for a system with 2 atoms, I've determined the minimum k-point mesh with success, but when I compute the Total energy varying the PWcE, it seems that the convergence...

First off, this is not a homework question. I am helping someone with an alternating series, and, for some reason, I am finding myself completely baffled by this one. I have an alternating series that takes the form: \sum\limits_{j=1}^\infty(-1)^j\frac{\sqrt{j}}{j+5} I know that the...

Probably is a silly question, but how could I prove that the function (expressed in polar coordinates) \left(\rho^4\cos^2{\theta} + \sin^3{\theta}\right)^{\frac{1}{3}} - \sin{\theta} converges to 0 as rho->0 uniformely in theta (if it is true, of course)?

Homework Statement Show convergence of s_{n+1}= \sqrt{2+\sqrt{s_n}} where s_1 = \sqrt{2} and that s_n<2 for all n=1,2,3... Homework Equations Let {p_n}be a sequence in metrice space X. {p_n} converges to p iff every neighborhood of p contains p_n for all but a finite number of n.The...

n! / nn I have proven that the sequence converges numerically, but I can't do it analytically, and can't do anything for the series (maybe it the series doesn't converge?)

Homework Statement "Determine whether the following series converge: \sum_{n \geq 2} \frac{n^{ln (n)}}{ln(n)^{n}} and \sum_{n \geq 2} \frac{1}{(ln(n))^{ln(n)}} Homework Equations The convergence/divergence tests (EXCEPT INTEGRAL TEST): Ratio Dyadic Comparison P-test...

When I use the calculation from Wikipedia that says that the radius of convergence of a series is lim as n goes to infinity of |an/an+1|, I get for the Taylor series expansion of ln(x) around a=2 the answer of an infinite radius of convergence, which would mean that it would be valid everywhere...

Hello was wondering if anyone could help me prove that: Suppose (sn) converges to s not equal to 0 and ( sntn) converges to L. Prove that (tn) converges

I have a question about the ratio test. Suppose it proves inconclusive, we must than use another test to check for conditional convergence - 1) this test has to be associated with an alternating series, such as the Alternating Series Test, correct? (we wouldn't be able to use something like...

Good day dear fellows. I am given the following series h(x) = \sum_{n=1}^{\infty} \frac{1}{x^2+n^2}. It is requested that I show that h(x) is continuous on R. I did the following: use the Weirerstrass M-test to show uniform convergence, and then, using the continuity of the functions...

Homework Statement i was doing this exercise and came across this example. ∞ Ʃ (x^n)/ln(n+1) n=1 The Attempt at a Solution i know you have to do the ratio test which is lim | a(n+1)/a(n)| n>∞ i got to lim | [x ln(n+1)] /ln(n+2) | n>∞ and have no idea how to continue? is...

Homework Statement Hi there, I have just started taylor series for my course.. most seems arlgiht so far, however when it comes to validating a given series( tayor or maclaruin), I have an idea on how to find out the x value.. but I don't know what I am doing wrong.Take for example: The...

Homework Statement I've attached the question Homework Equations The Attempt at a Solution I'm not exactly sure how to do this question. Is it an interval of convergence question where i simply let log(1+2x) < 1 and solve for x??

1. If {an} and {bn} are convergent, then {an士bn} and {anbn} are also convergent 2. If {an} and {bn} are convergent and there exists a constant k>0 such that |bn| > k for all n=1, 2, ..., then {an/bn} is also convergen

Homework Statement X, Y, (X_n)_{n>0} \text{ and } (Y_n)_{n>0} are random variables. Show that if X_n \xrightarrow{\text{P}} X and Y_n \xrightarrow{\text{P}} Y then X_n + Y_n \xrightarrow{\text{P}} X + Y Homework Equations If X_n \xrightarrow{\text{P}} X then...

Im struggling with the concept of this basic sequence question. Let x(n) be a sequence such that lim(n->00) (nx(n)) = 0 i.e. it converges to zero... How could i show that there is an N s.t. for all n≥N : -1 < nx(n) < 1 Any tips would be great.. I don't want an answer.. I want to...

I am given f_n(x)=\frac{nx}{nx+1} defined on [0,\infty) and I have that the function converges pointwise to 0 \ \mbox{if x=0 and} 1\ \mbox{otherwise} Is the function uniform convergent on [0,1] ? No. If we take x=1/n then Limit_{n\rightarrow\infty}|\frac{1/n*n}{1+1/n*n}-1|=0.5...

Homework Statement \sum_{n=1}^{\infty}(-1)^{n+1}\frac{\sqrt{n}+1}{n+1} Homework Equations absolute convergence test The Attempt at a Solution by book says that the series converges because \sum_{n=1}^{\infty}\frac{\sqrt{n}+1}{n+1} converges but they don't show how the absolute...

Homework Statement Find the minimum number required (value of n) for the average deviation of the Fourier Series to fall below 2% Homework Equations Use the Uniform Convergence of Fourier Series. Where Sm is the partial sum of the Fourier Series. C is constant. Here C is ∏^2 So...

Homework Statement Given each of the functions f below, describe the set of points at which the Fourier series converges to f. b) f(x) = abs(sqrt(x)) for x on [-pi, pi] with f(x+2pi)=f(x) Homework Equations Theorem: If f(x) is absolutely integrable, then its Fourier series converges to f...

Homework Statement I need to understand and prove the following: That if a>1 the function diverges, except for a special case x_0= b/(1-a). Then if a=-1 diverges for some cases and converges if x_0 is b/2. Again, not to clear on this. Homework Equations lim n →∞...

Homework Statement This is a homework question for a introductory course in analysis. given that a) the partial sums of f_n are uniformly bounded, b) g_1 \geq g_2 \geq ... \geq 0, c) g_n \rightarrow 0 uniformly, prove that \sum_{n=1}^{\infty} f_n g_n converges uniformly (the whole...

Homework Statement I need to calculate the point of divergence for this exponential function : F(x)= 5.282 * exp ( -0.01726 * x ) may you help me in finding the method to solve such problems ? Homework Equations The Attempt at a Solution

Homework Statement Consider the system x = \frac{1}{\sqrt{2}} * \sqrt{1+(x+y)^2} - 2/3 y = x = \frac{1}{\sqrt{2}} * \sqrt{1+(x-y)^2} - 2/3 Find a region D in the x,y-plane for which a fixed point iteration xn+1 = \frac{1}{\sqrt{2}} * \sqrt{1+(x_n + y_n)^2} - 2/3 yn+1 =...

Hello, I'm pondering over this research question. Let's suppose you've got a bunch of units which can be colored black or white. They're roaming around 2d grid in random walk. Any time a unit meets with another unit, it has an option to change color. It doesn't have to though, depending...

Hi i have to show that the series 1+2r+r2+2r3+r4+2r5+... converges for r=\frac{2}{3} and diverges for r=\frac{4}{3} using the nth root test. The sequence \sqrt[n]{a_{n}}comes a bit complicated so i was wondering if I can remove the 1st term a1=1 and show that 2r+r2+2r3+r4+2r5+... converges...

Homework Statement Use the definition of convergence to prove that lim n→∞ (1/2)^n=0 The definition of convergence says |a_n-L|<ε Homework Equations The Attempt at a Solution As I understand it: |(1/2)^n-0|<ε |(1/2)|^n<ε then I need to solve for n...

The problem statement Let f:[a,b]→\mathbb{R} be differentiable and assume that f(a)=0 and \left|f'(x)\right|\leq A\left|f(x)\right|, x\in [a,b]. Show that f(x)=0,x\in [a,b]. The attempt at a solution It was hinted at that the solution was partly as follows. Let a \leq x_0 \leq b. For all x\in...

So a well-known theorem from Lebesgue integration is the dominated convergence theorem. It talks about a sequence f_1,f_2,\ldots of functions converging pointwise to a function f. And if |f_n(x)| \leq g(x) for an integrable function g, then we have \int f_n \to \int f. But what if we have a...

Homework Statement The following is a modification of Newton's method: xn+1 = xn - f(xn) / g(xn) where g(xn) = (f(xn + f(xn)) - f(xn)) / f(xn) Homework Equations We are supposed to use the following method: let En = xn + p where p = root → xn = p + En Moreover, f(xn) = f(p + En) = f(p) +...

Homework Statement I am asked to comment on the convergence/divergence of three series based on some given information about a power series: \sum_{n=0}^{\infty}c_nx^n converges at x=-4 and diverges x=6. I won't ask for help on all of the series, so here's the first one...

Since I don't know how to use latex I have posed my question in word file. Yours help is greatly appreciated.

Homework Statement show that x_n converges to x if and only d(x_n, x) converges to 0. Homework Equations |x_n - x| < ε for all ε>0 The Attempt at a Solution well d(x_n,x) converges to 0 if d(x_n,x)<ε i just don't know how to relate that back to |x_n - x|

1. The problem statement: In what region can we choose x0 and get convergence to the root x = 0 for f(x) = e-1/x^2 Homework Equations xn+1 = xn - f(xn) / f'(xn) The Attempt at a Solution The only thing I've come across is a formula that says |root - initial point| < 1/M where M =...

If you are given part of a period of a Function, what rules would you apply to draw out the full function, so that it converges as quickly as possible as a Fourier series? thanks

Homework Statement Find the power series representation for the function f(x)=x/(x^2-3x+2) and determine the interval of convergence. Homework Equations The Attempt at a Solution First I separate into partial fractions 2/(x-2) - 1/(x-1) 2/(x-2) = sum n=0 to infinity (x/2)^n...

I was just watching a television program about gravity today and it got me wondering what gravity was exactly. Most analogies used to describe gravity are of a heavy ball on a bed sheet. The ball creates a depression in the sheet and objects placed on the sheet will fall in towards the ball...

Homework Statement I'm asked to specifically use the Ratio Test (formula below) to determine whether this series converges or diverges (if it converges, the value to which it converges is not needed.) \sum_{n=1}^{\infty}\frac{n}{(e^n)^2} Homework Equations Ratio Test: If a_n is a sequence...

Homework Statement prove that the series summation from n=3 to infinity of (1/(n*log(n)*(log(log(n))^p)) diverges if 0<p<=1 and converges for p>1. Homework Equations The Attempt at a Solution 2^n*a(2^n)= 1/(log(2^n)*(log(log(2^n))^p)). this is similar to the summation from n=2 to...

I know this is like very basic, but my brain just somehow couldn't accept it! Homework Statement I don't understand why does the sequence (rn) converges to 0 as n -> infinity when -1<|r|<1 The Attempt at a Solution i did quite a few ways to convince myself. Firstly, we know that...

Homework Statement assume summation of series An converges with all An>0. Prove summation of sqrt(An)/n converges Homework Equations The Attempt at a Solution I Tried using the ratio test which says if lim as n goes to infinity of |Bn+1/Bn|<1 then summation of Bn converges. I let Bn...

Okay, so I didn't really understand the professor when he talked about the speed of convergence of Fourier series. The question is what kind of functions converge faster than what kind of other functions using Fourier series representation. My guess from what I have absorbed is that functions...

Hi, Here's another question from my analsysi HW. I get that the two sequences are equal but I'm not sure how to write it out. Any help would be great. Thanks. Homework Statement Prove that a sequence {a_n} converges to 0 iff the sequence {\lvert a_n\rvert} converges to 0. Homework...

Hi, I'm doing some homework from my analysis class. I honestly have no idea where to start. Any help would be appreciated. Homework Statement Let {a_n} be a sequence that converges to 0, and let {b_n} be a sequence. Prove that the sequence a_n b_n converges to 0. Homework Equations...

1. Prove that n^(1/n) converges to 1. 3. I've attempted to define {a} = n^1/n - 1 and have shown, using the binomial formula, that n=(1+a)^2>=1+[n(n-1)/2]*a^2. I think I'm on the right track but don't know how to bring this back to the original problem to prove convergence even after staring...