Power series Definition and 629 Threads

Homework Statement Find the power-series expansion about the given point for the function; find the largest disc in which the series is valid. f(z) = z^3 + 6z^2-4z-3 about z0=1. Homework Equations The Attempt at a Solution The series is fine. Since it's a polynomial, there are only three...

Homework Statement Find a power series representation for the given function using termwise integration. f(x) = \int_{0}^{x} \frac{1-e^{-t^2}}{t^2} dt Homework Equations The Attempt at a Solution Well, I figured I could rewrite it like this using the Maclaurin series for...

Homework Statement Compute \sum_{n=0}^{\infty} p^n cos(3nx) for \abs{p} \textless 1 , where p \in \mathbb{R} . Homework Equations The Attempt at a Solution I was thinking that maybe this could be approached as a telescoping series, but I'm not really sure if it is. Would that...

I was going through http://mathworld.wolfram.com/LaguerreDifferentialEquation.html" in Wolfram which gives brief details about finding a power series solution of the Laguerre Differential Equation. I was reading the special case when v = 0. I read earlier from Differential Equations by Lomen...

Homework Statement Find a power series representation for f(x) using termwise integration, where f(x) = \int_{0}^{x} sin(t^3) dt . Homework Equations The Attempt at a Solution I've never done this before, but apparently, if I have a power series representation for sin(t^3), I...

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...

Homework Statement The function f(x)=\frac{1}{1+x^{9}} can be expanded in a power series \sum^{\infty}_{0} a_{n}x^{n} with center c = 0. Find the coefficient a_{27} of x^{27} in this power series. 2. The attempt at a solution I can get to: \sum^{\infty}_{0} (-1)^{n}(-x^{9})^{n} which I...

Homework Statement Give an example of a power series tha converges on the interval (3,6), but on no larger interval. Give some justification. Homework Equations The equation needed is probably that for a power series: \Sigma cn(x-a)n The Attempt at a Solution I'm not sure at all. The...

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...

Homework Statement If a_v > 0 and \sum_{v=0}^{\infty} a_v converges, then prove that \lim_{x \rightarrow 1^-} \sum_{v=0}^{\infty} a_v x^v = \sum_{v=0}^{\infty} a_v . Homework Equations The Attempt at a Solution Since \sum a_v converges, then we can say that \sum a_v x^v converges for x...

Find a power series sol'n: (x2-1)y'' + 3xy' + xy = 0 Homework Equations let y = \Sigma (from \infty to n=0) Cnxn let y' = \Sigma (from \infty to n=1) nCnxn-1 let y'' = \Sigma (from \infty to n=2) n(n-1)Cnxn-2 The Attempt at a Solution I wrote the differential eq as...

Hi, I have 2 questions regarding how to expand power series. 1). Find the power series expansion of Log z about the point z= i - 2 2). Expand the function 1/(z^2 + 1) in power series about infinity Any help will be greatly appreciated. This is because I am totally unsure about what...

Please can somebody help me with this problem y" + y' + sin^2(x)y - 2sinx = 0 I used power series method and i used the macclurin expresion for sinx but i was not able to get a recurrence formula.

I need to find \frac{1}{i+z} as a power series in z. I want to know if am doing this right. If i use the taylor series here by doing f(z) = z^i f'(z) = i z^{-1} z^i f''(z) = i (i-1) z^{-2} z^i This taylor series is just for z= i...

Homework Statement I'm having a little trouble concerning the part where we have to calculate the endpoints of a known interval of convergence in order to see whether they are convergent or divergent. In this case, is the summation of n=0 to infinity of (-1)^n/10 diverging or converging. Are...

Homework Statement I am confused about how to find a sum of a power series, especially when it contains factorials and I can't quite get it to look like a geometric series. Is it the same thing as finding a limit (and then I would follow the various tests for convergence of the different...

Hello, I need to find the sum(as a function of x) of the power series \Sigma^{\infty}_{n=0}\frac{(x+1)^n}{(n+2)!} The hint i was given was compare it to the Taylor series expansion of ex. Im not sure even how to start this problem and any help is much appreciated.

How to do expansion as power series of any random function?? The template doesn't really apply because this is a general question rather than a specific problem. If I am given some function, how do I expand it as a power series? For the past two semesters of my physics degree, I have been...

Homework Statement Hi all. In my book on complex analysis, they discuss complex power series. They use a variety of "tests" to determine absolute convergence, but they never say if this also implies convergence. Does it? Niles.

Which of the following expansions is impossible? a \sqrt{x-1} in powers of x b \sqrt{x+1} in powers of x c ln(x) in powers of (x-1) d tanx in powers of (x-\pi / 4 ) e ln(1-x) in powers of x What are htey asking and how do i do this? the answer is A by the way

Homework Statement Solve: y'' + y' - 2y = 0 y(0) = 1 y'(0) = -2 Homework Equations The Attempt at a Solution I found: http://image.cramster.com/answer-board/image/cramster-equation-200942414569633761817696516250247.gif So the recurrence relation is...

Homework Statement How do I prove a power series is onto? Since I cannot calculate directly, especially I haven't learned Jodarn Normal form. Homework Equations The Attempt at a Solution By showing 1-1, I tried ∑(1/n!)[(M)^n-(N)^n]=0, what can I conclude from this step?

Homework Statement In the following series': http://image.cramster.com/answer-board/image/cramster-equation-2009410014306337491927047975008434.gif According to my book, we only have a common range of summation here for n >= 2. Therefore we need to treat n = 0 and n = 1 separately...

Homework Statement Find the radius and interval of convergence of the given power series. \sum^{\infty}_{n=0}\stackrel{100^{n}(x+7)^{n}}{n!} Homework Equations Ratio Test The Attempt at a Solution My real question is: Can the radius be 0? Cuz that's what I get. Would the...

Homework Statement The problem just states to find the Laplace Transform of cos(kt) from its power series expansion, instead of using the formula for the transform of a periodic function.Homework Equations Equation for Laplace transform of a function f(t) ->\int(e^{-st}f(t))dt Power Series...

Homework Statement Suppose the real power series \sum ^{\infty}_{n=0}c_{n}x^{n} has radius of convergence R > 0. Define f:= \sum ^{\infty}_{n=0}c_{n}x^{n} on I:= (-R, R) and let b \in I. Show that there exists a power series \sum d_{n}(x-b)^{n} that converges to f(x) for |x-b| < r - |b|...

When solving diff-eq's given initial values, e.g. y'' - 2y ' + y = 0 y (0) = 0 y ' (0) = 1 Can one assume immediately that y(0) = c0 and y ' (0) = c1 ? Since these are the first 2 terms in the series? Thanks!

Homework Statement 1. Find a power series for (1-x)^(-1/2) 2. Find a power series for (1-x^2)^(-1/2) 3. Find a power series for arcsin(x) Homework Equations Binomial series, (1+x)^k= 1 + kx + k(k-1)x^2/2!+ k(k-1)(k-2)x^3/3!+... The Attempt at a Solution for 1., I have 1+ (-1/2)(-x) +...

Homework Statement Determine the series of the given function: f(x) = 10 / (1-5*x) Homework Equations Power series of 1/(1-x) = Σ from n=0 to n=infinity of (x^n) The Attempt at a Solution f(x) = 10/(1-5x) = 10*(1/1-5x) = 10 * Σ(5x)^n = 10 * Σ(5^n)*(x^n) = Σ (50^n)*(x^n) <--- Not sure if...

Homework Statement Word for word, the book says "Find a power series that represents 1/(1+x^2) on (-1, 1) Homework Equations It's in the chapter that talks about power series, so I think they want me to use the fact that 1/(1-x) is a power series with a=1 and r=x, but if I just...

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...

Homework Statement Solve (1-4x^2)y''+34x\cdot y'-70y=0 Homework Equations Basically, I found the recurrence relationship to be: a_{n+2}=\frac{2 (-7 + n) (-5 + 2 n)}{(n+1)(n+2)}a_n} Now, I solve for y1 where y1 had a_0=0 and a_1 = 1. It is a simple polynomial of degree 7...

How do I go about finishing/calculating this? Homework Statement Use a power series to approximate \int\cos 4x\log x dx to six decimal places. (bounds are from pi to 2pi) Homework Equations The Attempt at a Solution So I broke down the equation first: \int\cos 4x\log x dx =...

Homework Statement y"+3y'+2y= sin x y(0)=0 y'(0)=1 Evaluate y(0.1) Homework Equations Power Series Equation The Attempt at a Solution

Hey there! I'm new here and I just want to ask anyone willing how to solve this problem using power series: y"+3y'+2y= sin x y(0)=0 y'(0)=1 Evaluate y(0.1) Thanks! :smile:

Homework Statement [Directions to problem] Show that the function of x gives a power series expansion on some interval centered at the origin. Find the expansion and give its interval of validity. \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I have...

Homework Statement power series expansion of: ((cosh x)/(sinh x)) - (1/x) Homework Equations cosh x = (1/2)(ex + e-x) sinh x = (1/2)(ex - e-x) The Attempt at a Solution what i have so far: I simplified the first part of the eq to read : e2x-1 e2x-1 now I am stuck...

Homework Statement Find the closed form of the following power series 1+3x+6x^2+10x^3+15x^4+21x^5+... Homework Equations 1+x+x^2+.. = 1/(1-x) The Attempt at a Solution I tried differentiating but couldn't get it to any expression that I know the sum for.. I was playing around...

I have to find the "second smallest root" of the following equation : 1-x+(x^2)/(2!)^2-(x^3)/(3!)^2+(x^4)/(4!)^2+...=0 Matlab returns quite a satisfactory answer. >> p=[1/518400 -1/14400 1/1576 -1/36 1/4 -1 1] p = 0.0000 -0.0001 0.0006 -0.0278 0.2500 -1.0000 1.0000...

Homework Statement \begin{equation} 1 - x + \frac{x^2}{(2!)^2} - \frac{x^3}{(3!)^2} + \frac{x^4}{(4!)^2} +... = 0 \nonumber \end{equation} Homework Equations To find out the power series in the LHS of the given equation. The Attempt at a Solution I have tried to solve it by...

I have to find the power series f(x)= 3 / (2 -5x). I devided everything by 2 so I have (1/2) / (1 - 5x/2) = \sum ar^n and then through some steps I have 15x^n / 2^n+1 and I on the right track? How do I find the interval of convergence?

Hey all, So I have a physics final coming up and I have been reviewing series. I realized that I'm not quite sure on what the differences are between a Taylor series and a power series. From what I think is true, a taylor series is essentially a specific type of power series. Would it be...

Homework Statement x²y'' + (x² - 2)y = 0 Homework Equations N/A The Attempt at a Solution I divided through by x² to get the equation in standard form. Then I plugged in the power series representation for y and y'' into the equation and got to this point...

Homework Statement Problem: Find the value of b for which Homework Equations Power series for (1/1+x) or in this case, power series for (1/1+b) The Attempt at a Solution I keep getting ln (-5/6) as the answer, but apparently the correct answer is ln (5/6). I do not see why...

hey, this is my first time posting, my question is find the power series representation for x/(1-x)^2 I know the representation for 1/1-x is x^n so does that mean x/(1-x)^2 is x^n^2? could use some clarification please

I never took complex analysis in undergrad and always regretted it, so I'm working through the book Visual Complex Analysis on my own. Really enjoying it so far. Homework Statement Actually you can view the problem...

Hello, I have a power series, and a problem with an integration with it, I don't understand why I should integrate it from zero at one point. I have attached a detalied explanation of the problem. http://img79.imageshack.us/img79/5156/powerseriespt4.jpg Any help would be greatly appreciated.

Homework Statement Find the terms up to x^5 in the power series solution of the following equation y''=(1+x^{2})y Homework Equations Power series, sum from 0 to infinity y=\sum a_{n}x^{n} The Attempt at a Solution At first I just differentiated each term separately and...

I'm having a problem with estimating a function using a power series... the problem is Use the power series for f(x)= (5+x)^(1/3) to estimate 5.08^(1/3) correct to four decimal places. I found all the derivatives of f(x) but I'm not sure how to make it into a power series or what form to...

How do you multiply McLaurin functions by "regular" power series? For example: If y' = sum (1, inf.) n*a_n x^n-1 and cos(x) = sum(0, inf.) (-1^n x^2n) / (2n)!, how do you find the product? If y = sum(0, inf.) a_n x^n and sin(x) = sum(0, inf.) (-1^n x^(2n+1)) / (2n+1)!, how do you find the...