Radius of convergence Definition and 140 Threads

Homework Statement Find the radius of convergence of \sum n=1 to ∞ of [(-1)^(n-1) x^(n)/(n)] and give a formula for the value of the series at the right hand endpoint. Homework Equations The Attempt at a Solution Not really sure how to start this. I know I'm supposed to use the...

Homework Statement Determine a lower bound for the radius of convergence of series solutions about a) x_{0}=0 and b) x_{0}=2 for \left(1+x^{3}\right)y''+4xy'+y=0. Homework Equations N/A The Attempt at a Solution The zero of P\left(x\right)=\left(1+x^{2}\right) is -1. The...

Homework Statement "Find the radius of convergence of the power series for the following functions, expanded about the indicated point. 1 / (z - 1), about z = i. Homework Equations 1 / (1 - z) = 1 + z + z^2 + z^3 + z^4 + ... + Ratio Test: limsup sqrt(an)^k)^1/k The...

Homework Statement Find the centre and radius of convergence: \stackrel{\infty}{n=1}\sum n.(z+i\sqrt{2})^{n} Homework Equations 1) Ratio test \left|\frac{a_{n+1}}{a_{n}}\right|<1 2) Textbook uses \stackrel{lim}{n-> \infty}\left|\frac{a_{n}}{a_{n+1}}\right| The Attempt at a Solution using...

Homework Statement Find the radius of convergence of the series: ∞ ∑ n^-1.z^n n=1 Use the following lemma: ∞ ∞ If |z_1 - w| < |z_2 - w| and if ∑a_n.(z_2 - w)^n converges, then ∑a_n.(z_1 - w)^n also...

1. The problem statement: Show that the following series has a radius of convergence equal to exp\left(-p\right) Homework Equations For p real: \Sigma^{n=\infty}_{n=1}\left( \frac{n+p}{n}\right)^{n^{2}} z^{n} The Attempt at a Solution...

Homework Statement Suppose the series \sum_{n=0}^{\infty} a_n x^n has radius of convergence R and converges at x = R. Prove that \lim_{x \to R^{-}}\large( \sum_{n = 0}^{\infty} a_n x^n \large) = \sum_{n = 0}^{\infty} \large( \lim_{x \to R^{-}} a_n x^n \large) 2. Question For the case R...

quick help on this i seem to be missing some logic or process determine the lower bound the radius of convergence of series solutions about the given X0 (2+x^2)y''-xy'+4y=0 xo=0

When approximating a function with a Taylor series, I understand a series is centered around a given point a, and converges within a certain radius R. Say for a series with center a the interval of convergence is [a-R, a+R]. Does this imply that: 1. There also exists a Taylor series expansion...

Hey , I was wondering if anyone could help me out with this question regarding calculating the radius of convergence of the infinity series of (1/n!)x^(n!) This is my work First we consider when abs(x) < 1 then we have 0 <= abs(x^n!) <= abs(x^n) so we know that the series converges...

Homework Statement The coefficients of the power series the sum from n=0 to infinity of an (x-2)^n satisfy ao=5 and an= [(2n+1)/(3n-1)] an-1 for all n is greater than or equal to 1. The radius of convergence of the series is A) 0 B) 2/3 C) 3/2 D) 2 E) infinite Homework Equations...

Having a hard time with this one: E 1/n^x , have tried too use n^-x=e^(-x ln n) which in turn e^(...) = lim n->OO (1-(x ln n)/n)^n and then go on with finding the centre, but I feel I'm far far off. How to get it like E an(x-c)^n and use the more straight foreward path.

Hi everyone :smile: When determining the radius of convergence of a power series, when should I use the ratio (a[sub n+1] / a[sub n]) test versus the root (|a[sub n]|^(1/n)) test? I know that I'm supposed to use the ratio only when there are factorials, but other than that, are these tests...

Hi there - I'm trying to work out the radius of convergence of the series \sum_{n \geq 1} n^{\sqrt{n}}z^n and I'm not really sure where to get going - I've tried using the ratio test and got (not very far) with lim_{n \to \infty} | \frac{n^{\sqrt{n}}}{(n+1)^{\sqrt{n+1}}}|, and with the root...

Can anyone please give me an example of a real function that is indefinitely derivable at some point x=a, and whose Taylor series centered around that point only converges at that point? I've searched and searched but I can't come up with an example:P Thank you:)

Homework Statement What feature of the ODE explains your value for the radius of convergence of the series y2? y2 is a series which satisfies the ODE and I found that it converges for \abs{2x^2} < 1. Homework Equations y2=x-\frac{2}{3}x^2-\frac{4}{15}x^5+ \cdots ODE...

Homework Statement Find the radius of convergence of the Series: \sum_{i=1}^{\infty}\frac{(2n)!x^n}{(n!)^2} The attempt at a solution I used the Ratio Test but I always get L = |\frac{2x}{n+1}| The answer is 1/4. I think I am mistaking with factorial.

Hi. Not really a homework question. Just a doubt i would like to confirm. Is the radius of convergence of a power series always equal to the radius of convergence of it's primitive or when its differentiated? I have done a few examples and have noticed this. I am trying to understand this...

Homework Statement Find the radius of convergence for \Sigma \frac{nx^{2n}}{2^{n}} Homework Equations Ratio test The Attempt at a Solution I apply the ratio test to get \frac{(n+1)(x^{2})}{2n}. I let n approach infinity, to get \frac{1}{2}. So, this series converges when |x2|<1...

Hi please could you assist me: questions posted below:Assuming the function f is holomorphic in the disk \[D(0,1) = \{ z \in \mathbb{C}:|z| < 1\}\], prove that \[g(z) = \overline {f(\overline z )} \] is also holomorphic in D(0,1) and find its derivative? Find the radii of convergence of the...

[SOLVED] radius of convergence Homework Statement Let D be th region in the xy plane in which the series \sum_{k=1}^{\infty}\frac{(x+2y)^k}{k} converges. Describe D.Homework Equations The Attempt at a Solution By the ratio test, we find the radius of converge of the series in x+ 2y to be 1...

Homework Statement Suppose that \sumanxn has finite radius of convergence R and that an >= 0 for all n. Show that if the series converges at R, then it also converges at -R. Homework Equations The Attempt at a Solution Since the series converges at R, then I know that \sumanRn = M...

[SOLVED] Radius of Convergence Homework Statement 1/(1+x^2) = sum ( (-1)^k*x^(2k) ; 0 ; inf) - A integrating arctan (x) = sum ((-1)^k * x^(2k+1) / (2k+1) ; 0; inf) B I know A has radius of converge of 1, and I calculated B to be 2. My assignment solution says "Similarly, the...

I am looking for radius of convergence of this power series: \sum^{\infty}_{n=1}a_{n}x^{n}, where a_{n} is given below. a_{n} = (n!)^2/(2n)! I am looking for the lim sup of |a_n| and i am having trouble simplifying it. I know the radius of convergence is suppose to be 4, so the lim sup...

Problem Statement: Compute the Taylor Series for (1+x)^1/2 and find the radius of convergence Problem Solution: The Taylor Series expansion I get is T(x) = 1 + (0.5*x) - (0.25*x^2)/2 + (0.375*x^3)/3! - (0.9375*x^4)/4! +...-... So to get radius of convergence I have to find a...

Problem: Suppose that {a_{k}}^{\infty}_{k=0} is a bounded sequence of real numbers. Show that \suma_{k}x^{k} has a positive radius of convergence. Work: I have attempted to use the ratio test and failed. I am suspicious I can try the root test, but I am not sure how to work it...

Homework Statement Find the radius of convergence of (-1)^n(i^n)(n^2)(Z^n)/3^nThe Attempt at a Solution i have got to lZl i (n+1)^2/3n^2 but am unsure how to complete it...

[SOLVED] radius of convergence of an infinite summation Homework Statement find the radius of convergence of the series: \sum\frac{(-1)^k}{k^2 3^k}(z-\frac{1}{2})^{2k} Homework Equations the radius of convergence of a power series is given by \rho=\frac{1}{limsup |c_k|^{1/k}}...

Radius of convergence, interval of convergence Homework Statement Find the radius of convergence and the interval of convergence of the following series. a) \sum_{n=0}^\infty \frac{x^n}{(n^2)+1} c) \sum_{n=2}^\infty \frac{x^n}{ln(n)} e) \sum_{n=1}^\infty \frac{n!x^n}{n^2} f)...

Homework Statement Prove that the radius of convergence \rho of the power series \sumck (z-a)^k over all k, equals 1/R when ck is not 0 and you know that: |\frac{ck+1}{ck}|=R>0 Homework Equations I was planning on using that the radius of convergence is in this case: \rho=...

Homework Statement f(x) = x^4 / (2 - x^4). Specify radius of convergence. Homework Equations Power Series f`(x) = c2 + 2c2(x-a) + 3c3(x-a)^2 + ... = (infinity)sigma(n=1) [n * cn * (x-a)^(n-1)] The Attempt at a Solution I'm not sure what to do. Usually, most problems are like x^3 /...

[b]1. The radius of convergence of the power series the sum n=1 to infinity of (3x+4)^n / n is a 0 b 1/3 c 2/3 d 3/4 e 4/3 [b]2. the sum n=1 to infinity of (3x+4)^n / n [b]3. no idea do the ration test to get abs value 3x+4 < 1 ?

Could someone please help me out with the following? I need to determine the radius of convergence of the following series. It is exactly as given in the question. \sum\limits_{n = 0}^\infty {\left( {3 + \left( { - 1} \right)^n } \right)^n } z^n The suggestion is to use the...

Homework Statement Find the radius of convergence of the following series. \sum\limits_{k = 1}^\infty {2^k z^{k!} } Homework Equations The answer is given as R = 1 and the suggested method is to use the Cauchy-Hadamard criterion; R = \frac{1}{L},L = \lim \sup \left\{ {\left|...

is it possible for "R" (radius of convergence) to be negative? is it possible for "R" (radius of convergence) to be negative? for example: -|x|<1 and R=-1?

Homework Statement "Find the radius of convergence and interval of convergence of the series" \sum_{n=0}^\infty \frac{x^n}{n!} Homework Equations Ratio Test The Attempt at a Solution \lim{\substack{n\rightarrow \infty}} |x/n+1| (I can't seem to get the |x/n+1| to move up where it should be)...

I have the Maclaurin series for cos (x), is their a way to find its radius of convergence from that? ALSO Is there a trick to find the shorter version of the power series for the Maclaurin series, I can never seem to find it so instead of the long series with each term but like E summation (the...

I really need help with this exercise. Consider the power series \sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{2n+1}. for z\in\mathbb{C}. I need to answer the following questions: a) Is the series convergent for z = 1? This is easy; just plug in z = 1 and observe that the alternating...

Hi folks. I need to find the radius of convergence of this series: \sum_{k=0}^\infty \frac{(n!)^3z^{3n}}{(3n)!} The thing throwing me off is the z^{3n}. If the series was \sum_{k=0}^\infty \frac{(n!)^3z^n}{(3n)!} I can show it has radius of convergence of zero. But z^{3n} means its only...

I am given this series: \sum_{n=1}^\infty\frac{2n}{n^2+1}z^n. First I have to find the radius of convergence; I find R = 1. Then I have to show that the series is convergent, but not absolutely convergent, for z = -1, i.e. that the series \sum_{n=1}^\infty(-1)^n\frac{2n}{n^2+1} is...