Power series Definition and 629 Threads

How do I write a derivation of the cosine power series? (I understand and can derive it, but it takes much space and is disjointed! ; how do you write the shortest and fastest derivation for it--briefly and fluently??)

Here's our equation: \frac{d^2\psi}{du^2}+(\frac{\beta}{\alpha}-u^2)\psi=0 This is the SE for the simple harmonic oscillator. My text goes through an elaborate solution to this DE and ends up resorting to a power series solution, not for psi, but for H, where \psi=H(u)e^{-u^2/2}. The text...

can someone tell mehow to expand (1 - \frac{2n}{(n+1)^2})^2 into power series or at least direct me to the formula!

A couple of days ago one of my teachers mentioned (when discussing the completeness of spherical harmonics) that {1,x,x^2,x^3,...} forms an overcomplete basis for (a certain class of) functions. This implies that a power series expansion of a function is not unique. And you can for instance...

I'm trying to show that the quotient of two power series Sum(n=o, infinity)[an*z^n] and Sum(n=0, infinity)[bn*z^n] is the power series Sum(n=0, infinity)[cn*z^n] where c0=a0/b0 and b0cn= (an-Sum(k=0, infinity)[bk*c(n-k)]). Is there a way of showing this by (Sum[bn*z^n])(Sum[cn*z^n])=Sum[an*z^n]...

Evaluate the indefinite integral as a power series and find radius of convergence. (i don't know how to type the integral and summations signs, sorry) (integral sign) (x-tan^-1x)/x^3 dx. ( if you write this out it makes more sense) i was able to find the power series of tan^-1x =...

Evaluate the indefinite integral as a power series and find radius of convergence. (i don't know how to type the integral and summations signs, sorry) (integral sign) (x-tan^-1x)/x^3 dx. ( if you write this out it makes more sense) i was able to find the power series of tan^-1x =...

Hi, I'm trying to solve a differential equation and I'm supposed to obtain a recursion formula for the coefficients of the power series solution of the following equation: w'' + (1/(1+z^2)) w = 0. The term 1/(1+z^2) I recognize as a geometric series and can be expressed as sum of 0 to...

I'm having some difficulty with this question: Solve the given differential equation by means of a power series about the given point x0. Find the recurrence relation. Also find the first four terms in each of two lin. indep. solutions. If possible, find the general term in each solution...

I don't understand how to get the inteveral of convergence from a problem: for example: infinity _ \ / -(1)^n(x-2)^n/(n*4^n) _ n=1 I know you have to use ratio test and it comes out to: |x-2|/4 The textbook says the series converges when |x-2|<4. How did they come to this...

Problem y^{\prime} = x^2 y General Comments There must be some kind of flaw in my solution as I don't get to the same result as the one my book provides: y = c_0 \sum _{n=0} ^{\infty} \frac{x^{3n}}{3^n n!} = c_0 e^{x^3 / 3} \qquad \fbox{CORRECT ANSWER} Any help is highly...

I was wondering how to create a power series expansion for the function x^2 / (1+x^2)... I've tried using the geometric series, but somehow i got stuck. thanks.

This one involves differentiation and integration of a power series. I need to find the first derivative, second derivative, and integral of this. f(x)=\sum(\frac{x}{2})^n if \frac{d}{dx}f(x)=\sum na_n(x-c)^{n-1} I guess I'm having problems figuring out what a_n is. Shouldn't it...

Problem: (a) Expand f(x)=\frac{x+x^2}{\left( 1-x\right) ^3} as a power series. (b) Use part (a) to find the sum of the series \sum _{n=1} ^{\infty} \frac{n^2}{2^n} \hline Solution: (a) f(x)=\frac{x+x^2}{\left( 1-x\right) ^3} = \left( x + x^2 \right) \left[ 1...

I'm not really having problems following the directions and arriving at the correct values, but I'm not really sure what those values mean or what I'm really figuring out. R = radius of convergence c = center of the series Take this one for example. \sum\frac{(2x)^{2n}}{(2n)!}...

I need to obtain the sum of the following series \sum _{n=1} ^{\infty} \frac{n^2}{2^n} Well, with the aid of Mathematica, I get the answer, which is 6. What I'm trying to do now is work my way backwards from there. I need to express it through the geometric series \sum _{n=0}...

I used Mathematica to confirm that \sum _{n=0} ^{\infty} \left[ \frac{\left( -1 \right) ^n \left( 2n \right) x^{2n+1}}{n! \left( n+1 \right) ! 2^{2n+1}} \right] + \sum _{n=0} ^{\infty} \left[ \frac{\left( -1 \right) ^n \left( 2n \right) \left( 2n+1 \right) x^{2n+1}}{n! \left( n+1...

Hi there, I have a question regarding the differentiation of a power series. I understand what my book says about the index in a power series... \frac{d}{dx} \left[ \sum _{n=0} ^{\infty} c_n \left( x - a \right) ^n \right] = \sum _{n=1} ^{\infty} n c_n \left( x - a \right) ^{n-1}...

Can anyone help me with this? It's been really annoying me and I think I am just forgetting something: Using the series expansion for cosx in powers of x find the first four non-zero terms of the corresponding series for sec x. I get obviously that as secx is 1/cosx it is a case of...

I think I got pretty close to the answer to this problem. However, I just can't obtain the right approximation at the end. Please, help me find where I made a mistake. Thanks. -------------------------------------------------------------- Use a power series to approximate the definite...

I need to know how to get the first 6 terms of the power series for: f(x)=ln(x) centered at x=1. Thanks

After setting out in the sums and collecting the terms in x^j I'm left with a series of expressions in a_2, a_3 etc as I believe I'm supposed to. However my first expression reads 2a_{2}+2a_{1}+a_-_{1}=0 Now I'm told that y(o) = 1 and y'(o) = 0 I think this means that a_0 = 1 and...

I felt pretty chuffed after getting the power series solution for a simple first-order ODE (with MathNerd's help) and thought I'd have a go at solving a second-order ODE using the same method. Then I realized I didn't understand it as well as I'd thought... The differential I'm attempting to...

I am trying to find the power series solution of y' = x^2y but don't know how to arrive at the answer of y = a_0exp(x^3/3). [I know that it's an easily solved separable equation, I'm just trying to figure out how to find the power series solution] My solution so far: Assume y...

The problem I have is that I have to find a function from the power series: f(x)=sigma (from n=0 to infinite) (cn)x^n ... where in cn the n is subscript and then the statement is given cn+4=cn ... where again n+4 and n are subscripts. then they tell you to suppose a=c0, b=c1, c=c2...

Hi, This was a question on a past exam, and I'm very confused about how to do this. We're not allowed calculators, so I'm sure there must be some simple solution that I'm missing. It asks: Use power series to estimate \int_{0}^{1/2}\frac{1}{\sqrt{1+x^3}}dx within 10^-3. I started off...

the question is can you come up with a power series whose interval of convergance is the interval (0,1] that is 0 < x < = 1 ? how about (0,infinity)? Give an explicit series of explain why you can't. The first part of this question where they ask if a series can have an interval of (0,1). I...

Hello, can anyone help me on this one? I am finding problem pretty tricky indeed. I know the power series (or Taylor series) for cosx is 1-x^2/2!+x^4/4!-x^6/6!+... (It is convergent for all element x is a member of set R) Q) Write down the power series for cos(u^2) up to the u^24 term...

[SOLVED] power series expansion for Laplace transform We are to find the Taylor series about 0 of e^t, take the tranform of each term and sum if possible. So I know the expansion of e^t is 1+x/1!+x^2/2!... x^n/n! then taking the tranform, 1/s + (1/1!)(1!/s^2) +(1/2!)(2!/s3)... and so on then...