Power series Definition and 644 Threads

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...

Lets say we have this series: a0+ a1(x-k)^1 +a2(x-k)^2 +a3(x-k)^3 = s(x) If I integrate the series a theorem in the books says that I will get the antiderivate S(x)+C, but won't C allways be equal to zero?

Homework Statement Use the power series method to find the general solution to: y'' + 2xy' + 2y = 0 Homework Equations The Attempt at a Solution I let y(x)=a0 + a1x + a2x^2 +...+anx^n + ... Then i found out what y'(x) and y''(x) was. I then equated the constant terms with...

Hi all, I'm having trouble understading this problem I got for homework. They're actually two problems in one, (a) and (b)... Any comment you can give me for understaing this will REALLY help. This is due Thursday, so please if anyone knows anything about this, can you share it with me? Thank...

I was trying to expand a scalar function with a power series but it accepts a vector argument. Do I simply use the multivariable power series expansion with the components of the vector acting as the argument OR do I use the single variable power series and take the vector's magnitude in the...

I'm having trouble with a few homework problems, so here are the problems and my thoughts. Homework Statement Use the power series to evaluate the function f(x)= \frac{1}{\sqrt{1+x^4}}-cos(x^2) at x=0.01. Use the first two terms in the series to approximate the function, but estimate the...

Homework Statement Solve the differential equation f' = 2xf2 with the initial condition f(0)=1 in the following way: i) First, assume that there is a solution given by a power series f(x) = with a positive radius of convergence. SUbstitude this into the differential equation and figure...

I have defined a power series about the point a = 1 as, f\left(a\right)= \sum^{\infty}_{n=0}\left(-1\right)^{n}\left[-\frac{1}{\left(e-1\right)^{n}}+\frac{1}{\left(e-1\right)^{n+1}}\right]\left(a-1\right)^{n} The terms of the power series are correct for all n, except n = 0. I need n = 0 to...

Homework Statement expand the exponential term in the equation y=2[e^{x+(x²/2)}-1] as a power series Homework Equations on wikipedia I found this... http://img297.imageshack.us/img297/1088/15139862vw6.jpg The Attempt at a Solution Do I substitute x+(x²/2) as "x" in the above...

Homework Statement Show that if a function f(x) can be expressed as an infinite power series, then it has the form f(x) = f(x0) + \sum^{\infty}_{n = 1}\frac{f^{n}(x0)}{n!}(x - x0)^{} Homework Equations The Attempt at a Solution I know that for an infinite power series: =...

Why is the domain of arctan power series |x|<=1? i understand why a power series has the domain |x|<1, but why is the power series of arctan |x|<=1? thanks! also, why are domains of the taylor expansions of coshx and sinhx: for all x?

1. Use multiplication or division of power series to find the first 3 nonzero terms in the Maclaurin series of the function x/sinx. 2. the maclaurin series for sinx is \sum(-1)^{n}x^{2n+1}/(2n+1)! 3. I've tried to divide x/sinx substituting sinx for the maclaurin series but I seem...

Hi My confusion is about this power series. If derivative of a function (f(x)) is another function(g(x)) then, this holds for the series terms of the functions. My question is If one knows this derivation relation Just two equal labelled series terms of two other functions. And one...

Under what conditions does a function have a power series representation? I am looking for a theorem that says if a function satisfies these conditions then it has a power series representation. Or does all functions have a power series representation?

Homework Statement The function f(x) =ln(10 - x) is represented as a power series in the form f(x) = (sum from 0 to infinity) of c_{n}x^{n} Find the first few coefficients in the power series. The Attempt at a Solution I know how to find the coefficients in a normal looking...

Homework Statement Suppose sum(a_n*x^n) represents a power series with radius of convergence (-R, R). Is it true that the series sum(n*a_n*x^n) is convergent? Prove or give a counter example. Homework Equations The Attempt at a Solution Let b_n = n*a_n*x^n Using ratio test...

Homework Statement Find the power series representation of the following: f(x) = e^(sin(x)) Homework Equations I know this to be true: e^x = (Inf Sum, n=0) (x^n/n!) The Attempt at a Solution So, in substituting sin(x) for x, I get: e^(sin(x)) = 1 + sin(x)/1 + sin(x)/2 + sin(x)/6 + ...

Homework Statement Evaluate the indefinite integral as a power series and find the radius of convergence \int\frac{x-arctan(x)}{x^3} I have no idea where to start here. Should I just integrate it first?

Homework Statement Is there a way to check if a power series representation of a function is correct? Homework Equations The Attempt at a Solution

Homework Statement Find the interval of convergence of f'(x) f(x) Sum from n=1 to infinity [(x-5)^n*(-1)^n]/[n5^n] Homework Equations The Attempt at a Solution My problem is I am unsure how to take the derivative with the n's and x's should I treat n as a constant...

Homework Statement See the attached png Homework Equations See the attached png The Attempt at a Solution See the attached png

Homework Statement solve the initial value problem: x(2-x)y'' - 6(x-1)y' - 4y = 0 y(1)=1 y'(1) = 0 hint: since the initial condition is given at x_0 = 1 , it is best to write the solution as a series centered at x_0 = 1 . Homework Equations I have attempted the question, but...

Homework Statement solve the initial value problem: x(2-x)y'' - 6(x-1)y' - 4y = 0 y(1)=1 y'(1) = 0 hint: since the initial condition is given at x_0 = 1 , it is best to write the solution as a series centered at x_0 = 1 . Homework Equations I have attempted the question, but...

For those who don't know I'm writing a tutorial (https://www.physicsforums.com/showthread.php?t=139690") in the tutorials forum. I have come to the point of introducing Transcendental functions. I would like to introduce the exponential function first (via the Taylor series) and then present...

Hi In trying to calculate the following sum: \sum_{i=1}^n{i^2} I found the following expansions: \sum_{i=1}^p \sum_{j=0}^{i-1} (-1)^j(i-j)^p {n+p-i+1\choose n-i} {p+1\choose j} My question is: is there an easier or more intuitive way to compute the limit of the sum above?

[SOLVED] !Power Series Solution to a Diff EQ! Homework Statement Find the first 5 term of a Power series solution of y'+2xy=0 (1) Missed this class, so please bear with my attempt here.The Attempt at a Solution Assuming that y takes the form y=\sum_{n=0}^{\infty}c_nx^n...

Homework Statement Use differentiation to find a power series representation for f(x) = 1/ (1+x)^2Homework Equations geometric series sum = 1/(1+x) The Attempt at a Solution (1) I see that the function they gave is the derivative of 1/(1+x). (2) Therefore, (-1)*(d/dx)summation(x^n) =...

Homework Statement Mass of rod: M mass of ball: m Length of Rod: L distance between rod and ball: x GPE is zero at infinty The questiopn asks to take the GPE of the rod/ball system, using the Power Series Expansion for ln(1+x) . Homework Equations U = -GMm/r The Attempt at a...

Hi: I have 2 questions about Power Series. The 1st one is a h/w problem and the 2nd one is an example from a textbook which I am having difficulty to figure out. Problem1: Given a power series g(x) = sum(0 to inf) of x^i/i!. Determine interval of convergence and compute g'(x)...

Homework Statement Find a power series representation for the function and determine the radius of convergence. heres the problem: http://img301.imageshack.us/img301/4514/30437250jj2.png Homework Equations The Attempt at a Solution i believe the derivative of arctant =...

Homework Statement I've tried to apply the ratio test to a problem that is a power series. here's the problem as a pic: http://img152.imageshack.us/img152/2751/35685690oj3.png Homework Equations The Attempt at a Solution I've gotten so far as you can see in the pic, I've...

[SOLVED] power series and taylor Homework Statement Let f be a function defined by f(x)=\frac{1+c x^2}{1+x^2}, and let x be an element of R for c\neq1, find the taylor series around the point a, and find the radius of convergence of the taylor series Homework Equations for power series...

For summation from 1 to infinity of x^(n)/[2^(n)*n^(4)] - I get the radius is 2 by ration test, but how do I get the interval, just by pluggin in?

Homework Statement Find the first non-zero terms, the general term for the cycloid power series, and the interval of convergence for the cycloid power series. cycloid: y=-a+acos(theta) x=a(theta)-asin(theta) Homework Equations I know that f(a)+ f'(a)(x-a)+ f''(a)(x-a)^2... will...

:devil: Can you handle it? Find the first non-zero terms, the general term for the cycloid power series, and the interval of convergence for the cycloid power series. cycloid: y=-a+acos(theta) x=a(theta)-asin(theta)

Find a power series solution for each of the initial value problems below: (a) y' (x) = cos x^2, y(0) = 0 (b) y'' - xy=0, y(0)=1, y' (0) = 0 Does anybody have any advice for this? Thanks!

http://img233.imageshack.us/img233/9559/21808788yx1.jpg so we took the derivative of the series, which i understand. but why is it that when the new n position was changed, why didn't the 2n+1 change as well? and i know that the n position changed b/c if it rained 0, we would have had x to...

[SOLVED] Power series representation, I really need help! :-] please let me know if i did this correctly f(x)=\arctan{(\frac{x}{3})} f'(x)=\frac{\frac{1}{3}}{1-(-\frac{x^2}{3^2})} \frac{1}{3}\int[\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{3^{2n}}]dx...

Homework Statement The function f(x)=5xarctan(3x) is represented as a power series. Find the first few coefficients in the power series. Homework Equations The power series is represented in the form sum(Cn*x^n) The Attempt at a Solution I've attempted to write the function as an...

f(x)=4x/(7+x). Find the first few coefficients and radius of convergence sum (n=0 to infinity) CnX^n The Attempt at a Solution I set up the equation into the form of a power series and got: (-1)^n*(4)^n*(x/7)^(n+1) But that doesn't seem to be right because I can't get the...