Power series Definition and 643 Threads

In mathematics, a power series (in one variable) is an infinite series of the form

where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
In many situations c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 1⁄10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.

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  1. S

    Textbook question on power series

    My textbook has an example on multiplication of power series. " Multiply the geometric series x^n by itself to get a power series for 1/(1-x)^2 for |x|<1 " from this we get the c_{n}=n+1 O.K. I get that the coefficients are 1 for all n but why +1. Could someone please explain this to me...
  2. A

    How Do You Use Power Series to Solve y``-3y`+2y=0?

    for this problem, y``-3y`+2y=0 how do you solve it using the power series method?
  3. O

    Can You Help Me Find a Power Series?

    Hi I"m having truoble with fnding the power series of the following::frown: 1+(x^3)/3+(x^6)/18+(x^9)/162+... Can anyone give me a hand? Also, any hints in finding a power series? Thanks !
  4. G

    Power Series For Function of Operators

    Hi, I'm looking for a general power series for a function of F of n operators. As normal, the operators do not necessarily commute. My first guess was: F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i p^j + b_{ij}p^i x^j However, I don't think this is correct as it is possible to...
  5. B

    Analyzing a Power Series: Convergence and Radius of Convergence

    I really need help with this exercise. Consider the power series \sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{2n+1}. for z\in\mathbb{C}. I need to answer the following questions: a) Is the series convergent for z = 1? This is easy; just plug in z = 1 and observe that the alternating...
  6. C

    How to Show that Coefficients of a Power Series for tan(x) are 0 for Even n

    Hi Just had a question. Assuming tan(x) is given by a power series with coefficiants (An). How can it be shown that An = 0 whenever n is even. Thanks
  7. RadiationX

    Determining Value of Power Series: How & Why?

    How do you know what value a power series is centered at? for example this power series is centered at 0: \sum_{n=0}^\infty\frac{x^n}{n!} what makes it centered at zero? this one is centered at -1: \sum_{n=0}^\infty{(-1)}^n{(x+1)}^n the only thing i can discern is that...
  8. quasar987

    Solving the Paradox of 0^0 in Power Series Analysis

    If I evaluate the power serie y(x) = \sum_{n=0}^{\infty}a_nx^n at x = 0, the first term is a_0(0)^0. But 0^0 is undefined, is it not? How is this paradox solved?
  9. R

    What are the rules for using the limit comparison theorem in power series?

    Hi, I'm preparing for my calculus II final and I had a question about Power Series. I'm posting the link for the answer solution to a practice exam (it's in pdf form) and I will ask questions based on that. http://www.math.sunysb.edu/~daryl/prcsol3.pdf On problem 12 it asks for the fifth...
  10. O

    What is the interval of convergence for this power series?

    summation from n=1 to infinity (x-2)^n/(n*3^n). My teacher got -1 <= x<= 5 as the interval of convergence because he found that x-2/3<1 Using the ratio test i get (x-2)n/(n+1)3, consistantly. This is driving me wild. :smile:
  11. P

    Power Series Expansion of f(x) = \frac{10}{1+100*x^2}

    the function f(x) = \frac{10}{1+100*x^2} is represented as a power series f(x) = \sum_{n=0}^{\infty} C_nX^n Find the first few coefficients in the power series: C_0 = ____ C_1 = ____ C_2 = ____ C_3 = ____ C_4 = ____ well f(x) = \frac{10}{1+100*x^2} can be written as...
  12. T

    Power series solutions to diffeq

    I am suppose to find the power series solutions to some diffeqs. y'=xy The method is to assume that y=\sum_{n=0}^{\infty}{c_nx^n} is a solution to the diffeq.Then since we can differenitate term by term we have \frac{d}{dx} \left[ \sum_{n=0}^{\infty}{c_nx^n}...
  13. tandoorichicken

    Help Needed: Checking & Solving Power Series for f(x) = x*ln(1+x)

    Two parts to this problem. On the first part I need someone to check my work, and I need help on solving the second part. (a) Find a power series representation for f(x) = ln(1+x). \frac{df}{dx} = \frac{1}{1+x} = \frac{1}{1-(-x)} = 1-x+x^2-x^3+x^4-... \int_{n} \frac{1}{1+x} = \int_{n}...
  14. B

    How to Derive the Cosine Power Series Using Euler's Identity?

    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??)
  15. N

    Why Use Power Series for Schrödinger Equation Solutions?

    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...
  16. S

    Expanding Power Series: (1- $\frac{2n}{(n+1)^2})^2$

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

    Nonuniqueness of power series expansion

    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...
  18. S

    Proving the Division of Power Series: A Rigorous Approach

    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]...
  19. J

    Evaluating Indefinite Integral as Power Series: Find Radius of Convergence

    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 =...
  20. J

    Finding Power Series and Radius of Convergence for Indefinite Integral

    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 =...
  21. M

    Recursion formula for power series solution

    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...
  22. cepheid

    How can the given differential equation be solved using power series?

    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...
  23. N

    Power Series - Can someone me understand them?

    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...
  24. D

    Differential Equation w/ Power Series Solution

    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...
  25. Y

    How can I create a power series expansion for x^2 / (1+x^2)?

    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.
  26. K

    Understanding Power Series Differentiation and Integration

    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...
  27. D

    Just need a quick explanation (Power Series)

    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...
  28. K

    Understanding R, c, and x in Power Series

    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)!}...
  29. D

    Finding the Sum of a particular Power Series

    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}...
  30. D

    Another Question Regarding Power Series

    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...
  31. D

    Differentiation of a Power Series

    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}...
  32. F

    Power Series: Find First 4 Terms of Series Expansion for Sec x

    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...
  33. D

    Using a Power Series to Approximate a Definite Integral

    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...
  34. I

    NEED HELP with power series PLEASE

    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
  35. B

    How Do You Solve for Coefficients in a Power Series for a 2nd Order DE?

    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...
  36. L

    More power series: 2nd-order ODE

    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...
  37. L

    Power series solution for 1st-order ODE

    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...
  38. D

    What is the equation for f(x) using just a,b,c,d in a geometric series?

    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...
  39. A

    Estimating \int_{0}^{1/2}\frac{1}{\sqrt{1+x^3}}dx using Power Series

    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...
  40. M

    Exploring Power Series: Convergence Intervals and Limitations Explained

    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...
  41. M

    What Rules Are Used to Simplify the Recursive Formula for Large j?

    I've done a power series solution to a differential equation and got the recursive formula for the coefficients below. Now I am to evaluate it for large j and I don't get the answer in the book. I'm not sure what method they are using to get the answer although their answer makes sense...
  42. I

    What is the power series for cos(u^2) up to the u^24 term?

    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...
  43. W

    Power series expansion for Laplace transform

    [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...
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