Power series Definition and 629 Threads

Homework Statement (x^2)y' = y Homework Equations The Attempt at a Solution Plugging in series everywhere I get the equation \sum na_{n}x^{n+1} = \sum a_{n}x^{n}. I try to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer. I also...

Homework Statement Find a power series representation for the function f(x) = \frac{(x-1)}{(3-x)^2}^2, valid for every x with |x|<3Homework Equations The equation that I think would be useful is \frac{1}{1-x} = \sum_{n=0}^\infty x^n The Attempt at a Solution I began by just looking at the...

Given the function f(x) = 1/(3-x) it can be represented in a power series by 1/3∑(x/3)n but is there any restriction on saying ∑(x-2)n except for x = 2? In the first case, R = 3 but the second case, R = 1 and on different intervals (i.e. (-3,3) and the other is (1,3). I just simply used the...

Homework Statement $$y' = xy$$Homework Equations $$y = a_{0} + a_{1}x + a_{2}x^{2}+... = \sum\limits_{n=0}^∞ a_{n}x^{n}$$ $$xy = a_{0}x + a_{1}x^{2} + a_{2}x^{3}+... = \sum\limits_{n=0}^∞ a_{n}x^{n+1}$$ $$y' = a_{1} + 2a_{2}x + 3a_{3}x^{2}... = \sum\limits_{n=1}^∞ n a_{n}x^{n-1}$$The Attempt...

What's the difference between these three? I only know Taylor series and its variants which I suppose is called power series (but I'm not sure). In that you just approximate around a single point using derivatives. But what are formal powers series and asymptotic expansion? I did see...

Hey, everyone. I am trying to find the power series of secant from the known power series of cosin, but my answer does not match up with Wolfram and Wikipedia. I know: cos(\theta) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 + ... So, using the first two terms (assuming a small angle)...

Homework Statement Find a power series for f(x) = \frac{1}{\sqrt{4+x^{2}}}, at x=0. 2. The attempt at a solution I have looked up the Taylor series of \frac{1}{\sqrt{4+x^{2}}}, but I don't find any similarity with a power serie like \sum_{n\geq 0} a_{n} x^{n} I don't know how to start...

Homework Statement I have to find the power series representation for integral (1/x) dx Homework Equations ln (1+x) The Attempt at a Solution This is very similar to ln(1+x) but I don't know if this helps me. Is this ln(x) shifted one to the right? So maybe I can use what is...

Homework Statement Question. Did I do this OK? Homework Equations The Attempt at a Solution A_n = Ʃ e^(n^2) x^n from n = 1 to ∞ So I tried the root test. After you take the nth root you have x e^n so then I took the limit of this as n-->∞ and I got infinity. I then said OK...

Homework Statement Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of (x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0 Note that x=0 is an ordinary point. Homework Equations y(x)=\sumckxk (k=0 to ∞) y'(x)=\sum(kckxk-1) (k=1 to ∞) y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞) The Attempt at a Solution...

Homework Statement I am trying to find the recursion relation for the coefficients of the series around x=0 for the ODE: y'''+x^2y'+xy=0 The Attempt at a Solution Therefore letting: y=\sum_{m=0}^\infty y_mx^m \therefore y'=\sum_{m=1}^\infty my_mx^{m-1} \therefore...

In the context of my work (linear differential equations), it can not be zero. But why?

Homework Statement Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of (x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0 Note that x=0 is an ordinary point. Homework Equations y(x)=\sumckxk (k=0 to ∞) y'(x)=\sum(kckxk-1) (k=1 to ∞) y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞) The Attempt at a Solution...

Say I have a simple series like \Sigma^{∞}_{n=0} X^{n} When I differentiate this series the first term goes to 0 because it's a constant. Does that mean that I have to adjust the index of the series from n=0 to n=1? If I don't do it, the first term still goes to zero as n(x^(n-1)) when n=0...

Homework Statement https://scontent-a.xx.fbcdn.net/hphotos-ash3/1390611_10201748262844961_2141774184_n.jpg I need help with 7b. Theorem 3 = termwise differentiation and theorem 4 = termwise integration.Homework Equations The Attempt at a Solution I have no idea how differentiation or...

Hi, I'm trying to find the series representation of f(x)=\int_{0}^{x} \frac{e^{t}}{1+t}dt . I have found it ussing the Maclaurin series, differenciating multiple times and finding a pattern. But I think it must be an eassier way, using the power series of elementary functions. I know that...

Homework Statement Problem 5 on the attached Sheet here Homework Equations We studied the Power Series Method and how to calculate a linearly independent solution if one solution is already known. So we need to find one solution (probably) using the power series method and then using...

Homework Statement Using that \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n for |x|<1 and that f'(x) =\sum_{n=0}^{\infty} (n+1)a_{n+1}(x-x_0)^n , write \sum_{n=0}^{\infty} n^2x^n in closed form. Homework Equations The Attempt at a Solution In this series, a_n = n^2 and x_0 = 0 ...

Please refer to attached image. Hi, I'm a bit lost here with the first question. Unfortunately the online lecture covering this material isn't available due to their having been made some technical difficulties, and I find our textbook difficult to comprehend! My lecture notes are pretty...

Let F(z) be the anti-derivative of the function f(z) = cos(z^3) with F(0) = 0. Express F(z) as a power series around z=0, giving both the first 3 non-zero terms and the general (nth) term. Hey guys really struggling with this integration and how to then express this as a power series. Any...

Hey guys! Suppose you have a function f(x)=1/2-x which you need to express as a power series. I am familiar with the conventional way of solving its series form, which involves taking out 1/2 from f(x) and arriving with a rational function 1/1-(x/2) which is easy to express as a power series...

Homework Statement I cannot write out the equation clearly so I am attaching a file. Homework Equations The Attempt at a Solution sin x= x- x^3/3! + x^5/5! - x^7/7! + ... sinx/(x)= 1- x^2/3! + x^4/5! - x^6/7! - x^26/27! + ... (-1)^k x^(2k) / (2k+1)! = g^(2k)(0) x^(2k)/(2k)...

Homework Statement The problem is to solve: y''+ty'+e^{t}y=0, y(0)=0 and y'(0)=-1 Homework Equations The Attempt at a Solution My main issue is the following: I normally find the recursion relation, and then factor out the t^{whatever} and I know that the coefficient to this...

"[F]ind the power-series expansion about the given point for each of the functions; find the largest disc in which the series is valid. 10. ##e^{z}## about ##z_{o} = \pi i##" (Complex Variables, 2nd edition; Stephen D. Fisher, pg. 133)$$f(z) = e^{z} = e^{z-a} \cdot e^{a} = e^{a} \cdot \sum...

Hello, My question is about power series. In most of questions i can find points with ratio test. But when i check points i can't understand style of parenthesis. Is there easy way? For example:\sum(n^(3)*(x-5)^n) I found check points 4<x<6 How can i decide to the parenthesis will be...

At exam today I was to calculate an improper integral of a function f defined by a power series. The power series had radius of convergence r=1. Inside this radius you could of course integrate each term, i.e. symbologically: ∫Ʃ = Ʃ∫ The only problem is that the improper integral went from 0...

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...

(Was posted in general physics forum also) I am currently reading Roger Penrose’s “Road to Reality”. In section 4.3, Convergence of power series, he refers to the sum of the series: 1 + x2 + x4 + x6 + x8 + ... = 1/(1-x2) Of course, this is true for |x| < 1, beyond which the series...

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 Prove. Homework Equations arctan(x) = x - \displaystyle \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} for -1 < x < 1. The Attempt at a Solution \displaystyle \sum^{∞}_{n=1} \frac{x^{n+2}}{n+2}

I was reading a paper the other day that made the following claim, and provided no reference for the assertion. I would like to find a reference or figure out how to derive the asymptotic behavior myself. The situation is as follows: Suppose we have a function ##f(z)##, defined as a power...

Homework Statement Find the power series of f'(x), given f(x) = x2cos2(x) Homework Equations Correct me if I'm wrong The Attempt at a Solution Can I just take the derivative of the solution I got previously? If so, what's a good way to write the sequence out so I can easily...

Homework Statement Can anyone explain to me why the answer to this question is D?: http://puu.sh/2FoET.png The Attempt at a Solution I'm not really sure where to begin, except I know that the series is centered at 0. I was also thinking that the given x's was the Interval of...

Homework Statement For the power series representation of, f(x)=1+x1−x which is 1+2∑from n=1 to inf (x^n), Where does the added 1 in front come from? How do I get to this answer from ∑n=0 to inf (x^n)+∑n=0 to inf (x^(n+1)) Homework Equations The Attempt at a Solution I arrived at ∑n=0 to inf...

Homework Statement for what values of x does the series converge absolutely?Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{4^n * x^n}{n!}The Attempt at a Solution Ratio Test \displaystyle \frac{4^{n+1} * x^{n+1}}{n+1)!} * \frac{n!}{4^n * x^n} 4x * limit (n->inf) \displaystyle...

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...

Here is the question: Here is a link to the question: Find a power series... Calc Help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

Homework Statement An exercise from advanced calculus by taylor : Homework Equations The Attempt at a Solution (a) ##\int_{0}^{x} tan^{-1}(t) dt = \int_{0}^{x} \sum_{n=0}^{∞} (-1)^n \frac{t^{2n+1}}{2n+1} dt = \sum_{n=0}^{∞} \frac{(-1)^n}{2n+1} \int_{0}^{x} t^{2n+1} dt =...

I am working through the Griffiths QM text and I am getting caught up on some the process he uses to derive the wave functions and energy levels for the QHO, via Frobenius/Power series method. I understand that the Schrodinger equation get recast into a summation form over the coefficients...

Homework Statement Solve ##(1-x)y''+y=0## at the point ##x_0=0##. Use this solution to find a solution to ##xy''+y=0## around the point ##x_0=1##. Homework Equations The Attempt at a Solution ##(1-x)y''+y=0## ##(x-1)y''=y## ##\displaystyle\sum_{k=2}^\infty a_k k...

hello Please see attachment which is a snapshot from MTW first page of chapter 19. Can someone please elaborate on how the equations 19.3b and c can be explained ? I know that equation 19.3a is a familiar formula but not so much the other two. I'm just confused. Thank you, Long live & clear...

Homework Statement Given x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+... = k, are there any values of x for the values k = -100, \frac{1}{2}, 100? Homework Equations The Attempt at a Solution I started by finding that the series is \Sigma^{\infty}_{n=1} \frac{(-1)^{n+1}x^n}{n} = k...

Homework Statement write a power series representation of the following: \frac{x}{15x^2 +1} Homework Equations the formula \frac{1}{1-x} = 1 + x + x^2 + ... = \sum_{n=0}^{∞} x^n The Attempt at a Solution we can rewrite the summnd like \frac{x}{15} \left(...

(Working out of Boas chapter 12, section 11) 3xy'' + (3x + 1)y' + y = 0 I'm asked to solve the differential equation using the method of Frobenius but I'm finding the way Boas introduces/explains/exemplifies the method to be incredibly confusing. So, I used some google-fu and was even...

1. The problem statement: Show that if the operator relation e^(ipa/ħ)xe^(-ipa/ħ) = x+a holds. The operator e^A is defined to the ∞ e^A= Ʃ(A^n)/n! n=0 [Hint: Calculate e^(ipa/ħ)xe^(-ipa/ħ)f(p) where f(p)is any function of p, and use the representation x=iħd/dp]...

Edit: Nevermind, figured it out. Thank you for readingOriginal problem: Find the interval of convergence \sum∞n=1 xn / n * √(n) * 3n Ratio Test, right? an+1/a I get to here and I can't figure out how to get rid of the ns: lim n→∞ abs(x/3)* [n*√(n) / (n+1)*√(n+1)] Solution, They break apart...

Show that, \[\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots\]

Homework Statement I am trying to find the sum of the series in the attachment. Homework Equations The Attempt at a Solution I have tried to use various series and their derivatives, to not much avail. I am not sure how to handle the n^2 factor. Should I break it down to two...

Hi, Homework Statement I am asked to prove that if the power series Ʃ(1,infinity) a_n(x-x0)^n converges at a point d, then it converges for every x that satisfies |x-x0|<|d-x0|. Homework Equations The Attempt at a Solution Obviously |d-x0|<r, where r denotes the radius of...

Homework Statement Show e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n Homework Equations J_k(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{(n+k)!n!}(\frac{x}{2})^{2n+k} The Attempt at a Solution Power series product (\sum^{\infty}_{n=0}a_n)\cdot (\sum^{\infty}_{n=0}...