Power series Definition and 629 Threads

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...

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?

: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...

1. Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence. f(x)= (7x-1)/(3x^2 +2x-1) 2. using the fact that 1/(1-x) =\sum from \infty to n=0 of x^n and the interval for convergence of that is (-1,1) 3. I know that...

Homework Statement Find the radius and interval of convergence for the following power series. \sum_{n = 2}^{\infty}\frac {(1 + 2cos\frac {\pi n}{4})^n}{lnn}x^n The Attempt at a Solution R = \frac {1}{\lim_{n\rightarrow\infty}\sqrt [n]{\frac {(1 + 2cos\frac {\pi n}{4})^n}{lnn}}}...

I have learned that if a function of one real variable can be defined as a power series, then this one is its Taylor series. Does the same occur with functions of 2 real variables? I mean, if a function f(x, y) can be defined as a power series, does this series is the Taylor series of f(x...

Homework Statement The function f(x)=10xarctan(5x) is represented as a power series http://img464.imageshack.us/img464/4131/formub5.jpg Find the first few(5) coefficients in the power series. Homework Equations I already know that the representation of arctanx is summation from...

For the fun of it, my DE book threw in a couple of problems involving nonhomogenous second order DE's in the section I'm currently going through. Although I have solved for the complementary solution, any suggestions on how to find the particular solution? For example, the one I'm looking at...

how could i expand something such as arctan'x into a power series. also how would you be able to find the power series for it?so far i have managed to work out that: arctan'x = \frac{1}{1 + x^2} \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 +...+ (- 1)^n x^{2n} how do you work out the radius of...

1. I this from my homework solution. (1-t/s)^n = exp(-t/s) as n goes to infinity I don't understand. I checked the exponential power series. It should be : exp(x) = summation (x^n / n!) n=0 to infinity How come it could be a exponential function ? 2. another is that...

y''+(x^2)y = 0 I tried to solve this problem using Power Series.But i can't make the solution in the form of series that have only two constants(a0,a1)that is, there are a0,a1, a2, a3. So i just wonder how can i make it has two constants.

Hi I stumbled across a power series pattern while working on a C algorithm and was wondering if this "discovery" has a name/reference anywhere. Basically, Here's what I found: for p = 1 the minimum number of terms, on the left side, satisfying the following is 1 a^p + b^p + c^p ... =...

I'm trying to do the question attached. I got the first three answers correct knowing that the nth derivative of a function evaluated at 0 divided by n! = c_n. However, I did the same for the others and the answer is incorrect. I know that I need the power series representation of that function...

My exam is coming up, I have 2 questions on infinite series. Any help is appreciated!:smile: Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG For part a, I got: g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)] For part b, I got: x ∫ tan^-1...

Homework Statement Start with the power series representation 1/(1-x) = sum from n=0 to inf. of x^n for abs(x) < 1 to find a power series representation for f(x) and determine the radius of convergence. f(x)=ln(5+x^2) Homework Equations The Attempt at a Solution Okay, so I...

Homework Statement Find a power series representation for the function f(x) = x / (4+x) and determine the interval of convergence. I have no idea how to begin this problem. My only guess would be trying to divide something out in order to simplify to something that I'm able to...