Radius of convergence Definition and 138 Threads

Greetings! I have a problem with the solution of that exercice I don´t agree with it because if i choose to factorise with 6^n instead of 2^n will get 5/6 instead thank you!

Greetings I have some problems finding the correct result My solution: I puted Y=6x+16 so now will try to find the raduis of convergence of Y so let's calculate the raduis criteria of convergence: We know that Y=6x+16 Conseqyently -21/6<=x<=-11/6 so the raduis must be 5/3. But this is not...

Good day I'm trying to find the radius of this serie, and here is the solution I just have problem understanding why 2^(n/2) is little o of 3^(n/3) ? many thanks in advance Best regards!

##\sum_{k=0}^\infty \frac {2^n+3^n}{4^n+5^n} x^n## in order to find the radius of convergence i apply the root test, that is ##\lim_{n \rightarrow +\infty} \sqrt [n]\frac {2^n+3^n}{4^n+5^n}## ##\lim_{n \rightarrow +\infty} \left(\frac {2^n+3^n}{4^n+5^n}\right)^\left(\frac 1 n\right)=\lim_{n...

Find the radius of convergence and interval of convergence of the series. $$\sum_{n=1}^{\infty}\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$ (1) $$a_n=\dfrac{(-1)^n x^n}{\sqrt[3]{n}}$$ (2) $$\left|\dfrac{a_{a+1}}{a_n}\right| =\left|\dfrac{(-1)^{n+1} x^{n+1}}{\sqrt[3]{n+1}}...

f(x) = 4x/(x-3)^2 Find the first five non-zero terms of power series representation centered at x = 0. Also find the radius of convergence.

Homework Statement This is from a complex analysis course: Find radius of convergence of $$\sum_{}^{} (log(n+1) - log (n)) z^n$$ Homework Equations I usually use the root test or with the limit of ##\frac {a_{n+1}}{a_n}## The Attempt at a Solution My first reaction is that this sum looks...

Hi, I've computed 512 terms of a power series numerically. Below are the first 20 terms. $$ \begin{align*} w(z)&=0.182456 -0.00505418 z+0.323581 z^2-0.708205 z^3-0.861668 z^4+0.83326 z^5+0.994182 z^6 \\ &-1.18398 z^7-0.849919 z^8+2.58123 z^9-0.487307 z^{10}-7.57713 z^{11}+3.91376 z^{12}\\...

Homework Statement in title Homework EquationsThe Attempt at a Solution so i know that i have to use the ratio test but i just got completely stuck ((2x)n+1/(n+1)) / ((2x)n) / n ) ((2x)n+1 * n) / ((2x)n) * ( n+1) ) ((2x)n*(n)) / ((2x)1) * (n+1) ) now i take the limit at inf? i am stuck here i...

Homework Statement ##f(x)=\sum_{n=0}^\infty x^n## ##g(x)=\sum_{n=253}^\infty x^n## The radius of convergence of both is 1. ## \lim_{N \rightarrow +\infty} \sum_{n=0}^N x^n - \sum_{n=253}^N x^n## 2. The attempt at a solution I got: ## \frac {x^{253}} {x-1}+\frac 1 {1-x}## for ##|x| \lt 1##...

Homework Statement ∞ ∑ = ((n-2)2)/n2 n=1 Homework Equations The ratio test/interval of convergence The Attempt at a Solution **NOTE this is a bonus homework and I've only had internet tutorials regarding the ratio test/interval of convergence so bear with me) lim ((n-1)n+1)/(n+1)n+1 *...

$\tiny{10.7.37}$ $\displaystyle\sum_{n=1}^{\infty} \frac{6\cdot 12 \cdot 18 \cdots 6n}{n!} x^n$ find the radius of convergence I put 6 but that wasn't the answer

Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...

Hey! :o I want to find for the following series the radius of convergence and the set of $x\in \mathbb{R}$ in which the series converges. $\displaystyle{\sum_{n=0}^{\infty}\frac{n}{2^n}x^{n^2}}$ $\displaystyle{\sum_{n=0}^{\infty}\frac{1}{(4+(-1)^n)^{3n}}(x-1)^{3n}}$ I have done the...

Supposing I have this expression: $$\sum_{n = 1}^{\infty} \frac{x^n}{3^n}$$ and I need to find the values for x for which this converges and the radius of convergence. I can use the radius test: $$\lim_{{n}\to{\infty}} |\frac{{x}^{(n + 1)} 3^n}{{3}^{(n + 1)} x^n}|$$ and this equals...

Hello, I have two questions regarding the Radius of convergence. 1. What should we do at the interval (R-eps, R) 2. It used definition to prove radius of convergence, but I am not sure if it is necessary-sufficient condition of convergence. I get that this can be a sufficient condition but not...

Homework Statement Let f(x)= (1+x)4/3 - In this question we are studying the Taylor series for f(x) about x=2. This assignment begins by having us find the first 6 terms in this Taylor series. For time, I will omit them; however, let's note that as we continuously take the derivative of this...

< Mentor Note -- thread moved to HH from the technical physics forums, so no HH Template is shown >[/color] How would you find the radius of convergence for the taylor expansion of: \begin{equation} f(z)=\frac{e^z}{(z-1)(z+1)(z-3)(z-2)} \end{equation} I was thinking that you would just...

Homework Statement Find all values of x such that the given series would converge Σ6n(x-5)n(n+1)/(n+11) Homework EquationsThe Attempt at a Solution By doing the ratio test I found that lim 6n(x-5)n(n+1)/(n+11) * (n+12)/[6n+1(x-5)n+1(n+2)] n→inf equals 1/(6(x-5)) * lim...

Homework Statement Hi everybody! I'm a little struggling to fully understand the idea of radius of convergence of a function, can somebody help me a little? Are some examples I found in old exams at my university: Calculate the radius of convergence of the following power series: a)...

Radius of convergence of $\displaystyle \sum_{j=0}^{\infty} \frac{z^{2j}}{2^j}$. If I let $z^2 = x$ I get a series whose radius of convergence is $2$ (by the ratio test). How do I get from this that the original series has a radius of convergence equal to $\sqrt{2}$?

Homework Statement Given the power serie ##\sum_{n\ge 0} a_n z^n##, with radius of convergence ##R##, if there exists a complex number ##z_0## such that the the serie is semi-convergent at ##z_0##, show that ##R = |z_0|##. Homework EquationsThe Attempt at a Solution Firstly, since...

Homework Statement Let ##\sum^{\infty}_{n=0} a_n(z-a)^n## be a real or complex power series and set ##\alpha = \limsup\limits_{n\rightarrow\infty} |a_n|^{\frac{1}{n}}##. If ##\alpha = \infty## then the convergence radius ##R=0##, else ##R## is given by ##R = \frac{1}{\alpha}##, where...

Homework Statement Σ(n=0 to ∞) ((20)(-1)^n(x^(3n))/8^(n+1) Homework Equations Ratio test for Power Series: ρ=lim(n->∞) a_(n+1)/a_n The Attempt at a Solution I tried the ratio test for Power Series and it went like this: ρ=lim(n->∞) (|x|^(3n+1)*8^(n+1))/(|x|^(3n)*8^(n+2)) =20|x|/8 lim(n->∞)...

Hi everyone, I am trying to evaluate the radius of convergence for the following power series: (k!(x-1)k)/((2k)(kk)) I have begun by trying to compute L = lim k-->inf (an+1/an). To then be able to say R = 1/L. So far i have L = lim k--> inf (kk(k+1)!)/(2(k+1)k+1k!) From here i am having...

I am attempting to evaluate the radius of convergence for a series that goes from k=0 to infinity. The series is given by (k*x^k)/(3^k). I have begun by using the ratio test and have gotten to the point L = (k+1)*x/3k Now i know i can find out the radius of convergence by simply saying R =...

Hello! (Wave) $$e^x= \sum_{n=0}^{\infty} \frac{x^n}{n!} \forall x \in \mathbb{R}$$ i.e. the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$. Could you explain me how we deduce that the radius of convergence of $\sum_{n=0}^{\infty} \frac{x^n}{n!}$ is $+\infty$? Do...

Homework Statement z ∈ ℂ What is the radius of convergence of (n=0 to ∞) Σ anzn? Homework Equations I used the Cauchy-Hardamard Theorem and found the lim sup of the convergent subsequences. a_n = \frac{n+(-1)^n}{n^2} limn→∞ |an|1/n The Attempt at a Solution I think that the radius of...

Homework Statement ∑ x2n / n! The limits of the sum go from n = 0 to n = infinity Homework EquationsThe Attempt at a Solution So I take the limit as n approaches infinity of aa+1 / an. So that gives me: ((x2n+2) * (n!)) / ((x2n) * (n + 1)!) Canceling everything out gives me x2 / (n + 1)...

Hi, I am likely just missing something fundamental here, but I recently just revisited series and am looking over some notes. In my notes, I have written that if ## \lim_{x \to +\infty} \frac{a_{n+1}}{a_n} = L ## Then ## | x - x_o | = 1/L ## But shouldn't the correct expression be $$ | x -...

Find the radius of convergence of the power series $\displaystyle\sum_{n=1}^{\infty}\dfrac{(-1)^{n+1}(z-1)^n}{n+1}$, $z\in \mathbb{C}.$

find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence. i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?

Hello. How do I find the radius of convergence for this problem? ##\alpha## is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$ I understand that we can use the ratio test to find R. And by using ratio test, I got R=1. But in the...

Hello. How do I find the radius of convergence for this problem? $\alpha$ is a real number that is not 0. $$f(z)=1+\sum_{n=1}^{\infty}\alpha(\alpha-1)...(\alpha-n+1)\frac{z^n}{n!}$$

Hello. I am stuck on this question. I'd appreciate if anyone could help me on how to do this. The question: Expand the following into maclaurin series and find its radius of convergence. $$\frac{2-z}{(1-z)^2}$$ I know that we can use geometric series as geometric series is generally...

Hello. I need explanation on why the answer for this problem is R = ∞. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$...

Hello. I need explanation on why the answer for this problem is $R=\infty$. Here's the question and the solution. Expand the function into maclaurin series and find the radius of convergence. $zsin(z^2)$ Solution: $$zsin(z^2)=z\sum_{n=0}^{\infty}(-1)^n\frac{z^{2(2n+1)}}{(2n+1)!}$$ Divide...

Homework Statement Series: \sum_{n=1}^{\infty}(-1)^{(n+1)}\frac{(x)^n}{na^n} what is the behaviour of the series at radius of convergence \rho_o=-z ? Homework Equations The Attempt at a Solution So I can specify that the series is monatonic if z is non negative as...

Homework Statement determine the radius of convergence of the series expansion of log(a + x) around x = 0 Homework Equations The Attempt at a Solution So after applying the Taylor series expansion about x=0 we get log(a) + SUM[(-1)^n x^n/(n a^n)] I understand how to get the...

Homework Statement I've found that the typical way for using ratio test is to find the limit of an+1/an However, my tutor said that radius of convergence can be found by finding the limit of an/an+1 and the x term is excluded. For example:Finding the interval of convergence of n!xn/nn my...

x^n/(2n-1) is the series. It starts at 1 and goes to infinity. I did the ratio test on it and got abs.(x) So the radius of convergence=1, and then I plugged -1 and 1 into the original series and got that they both converged. But the answer is [-1,1). Why aren't they both hard brackets?

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

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.

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...

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 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...

Homework Statement Suppose c_n is the digit in the nth place of the decimal expansion of 2^1/2. Prove that the radius of convergence of \sum{c_n x^n} is equal to 1. Homework Equations The Attempt at a Solution What I want to show is that limsup |c_n|^1/n = 1. Clearly for any...

1. Determine the raius of convergence and interval of convergence of the power series \sum from n=1 to \infty (3+(-1)n)nxn. 2. Usually when finding the radius of convergence of a power series I start off by using the ratio test: limn\rightarrow∞|((3+(-1)n+1)n+1xn+1/ (3+(-1)n)nxn| But...

Homework Statement Ʃ (from n=1 to ∞) (4x-1)^2n / (n^2) Find the radius and interval of convergenceThe Attempt at a Solution I managed to do the ratio test and get to this point: | (4x-1)^2 |< 1 But now what? How do you get the radius and interval? Any help will be appreciated! Thanks

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...