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. A

    Radius of convergence (power series) problem

    Homework Statement Ʃ (from n=1 to ∞) (4x-1)^2n / (n^2) Find the radius and interval of convergenceThe Attempt at a Solution I managed to do the ratio test and get to this point: | (4x-1)^2 |< 1 But now what? How do you get the radius and interval? Any help will be appreciated! Thanks
  2. C

    How Do You Determine the Radius and Interval of Convergence for a Power Series?

    I need help finishing this problem I am stuck. find radius of conv. and interval of convergence of the series. Ʃ k=0->∞ (1/k+1) (x)^k I have found all the way up to row=1 there for it is between (0,∞) so now that means that if absolute value of x<1 it converges if >1 it diverges but I...
  3. T

    Power Series for Volume of Balls in Riemannian Manifold

    I'm trying to work out the following problem: Find the first two terms of the power series expansion for the volume of a ball of radius r centered at p in a Riemannian Manifold, M with dimension n. We are given that Vol(B_r(p)) = \int_S \int_0^r \det(d(exp_p)_{tv})t^{n-1}\mathrm{d}t...
  4. A

    Representation of function as power series

    Homework Statement Represent 5x / (10 + x) as a power series, find c0, c1, c2, c3 and c4, find the radius of convergence The Attempt at a Solution I think I got the representation fine: Ʃ from 0 to ∞ = 1/2 [ (-1^n)(x^n+1) / 10^n ] Radius of Conv. = 10 But what the hell are those...
  5. H

    Nonlinear diff equation and power series

    Homework Statement i need to solve this diff equation. y' = x2 + y2 y = 1 when x = 0 i can assume that the answer is a power series on the form Ʃanxn andi only need the 4 first non zero terms of the power-series answer Homework Equations Ʃanxn The Attempt at a Solution...
  6. X

    Power Series to solve Second order Differential Equations

    Homework Statement When solving a D.E. with power series, I've encountered something along the lines of: (2 - r)^{2}g'' = -2 Homework Equations Power Series The Attempt at a Solution I know I am just supposed to assume such a series exists, and work from there. But I'm really...
  7. C

    Why Does Using Power Series Help Approach the Classical Limit in Physics?

    Homework Statement I have to show that the Planck radiation formula reduces to the Rayleigh-Jeans formula in the classical limit for blackbodies. The Attempt at a Solution I can easily show it using power series expansion of e^{(hc/\lambda kT)} but I don't understand really why using a...
  8. M

    Use ratio test to find radius and interval of convergence of power series

    Homework Statement Use the ratio test to find the radius of convergence and the interval of convergence of the power series: [[Shown in attachment]] Homework Equations an+1/an=k Radius of convergence = 1/k Interval of convergence: | x-a |∠ R The Attempt at a Solution I...
  9. B

    Finding A Function From A Power Series

    Homework Statement Find the sum of the series and its radius of convergence: \sum_{n=1}^{\infty} (-1)^{n+1}\frac{(x-1)^n}{n} Homework Equations The Attempt at a Solution I found the radius of convergence, but I wasn't sure how to find the sum of the power series.
  10. B

    What is the Interval of Convergence for the Power Series of f(x) = 2/(1 - x^2)?

    Homework Statement I am asked to find a power series for the function f(x) = 2/(1 - x^2), centered at 0. Homework Equations The Attempt at a Solution The only part I can't determine is the interval of convergence. I get stuck on the step |x2| < 1. What am I to do next?
  11. A

    How Do You Rewrite a Power Series for Radius of Convergence?

    If we have the series \sum_{n=0}^{\infty} 2^{-n}x^{3n}. We need to calculate the radius of convergence...the textbook says that, since the power of x is 3n instead of 3, we need to rewrite it in the form \sum_{n=0}^{\infty} a^{n}x^{n}..and we say that a_{3k}=2^{-k}. I'm not sure if I understand...
  12. D

    Can there be multiple power series representations for a function?

    I guess this is a simple question. Say I am tasked with finding the Taylor series for a given function. Well say that the function is analytic and so we know there's a taylor series representation for it. Am I gauranteed that this representation is the only power series representation for it...
  13. M

    Uniform Convergence of Power Series

    Given a power series \sum a_n x^n with radius of convergence R, it seems that the series converges uniformly on any compact set contained in the disc of radius R. This might be a silly question, but what's an example of a power series that doesn't actually also converge uniformly on the whole...
  14. M

    Criteria for a power series representation?

    I've used many different power series representations of functions and seem to always take it for granted that functions which are "nice" and continuous have such a representation. But what is the criteria for a function to have a power series representation? I know of some that don't, but...
  15. V

    Power Series Expansion: Find Alternatives to Ʃ ((-1)^(i-1))/i

    I posted this in the homework section, but it's not a homework problem. I basically need to know if the series Ʃ ((-1)^(i-1))/i can be represented in other ways (e.g. a Taylor series, but I doubt it). I know it converges to ln2, but I need to know if there's a series like x^2, x^4, ... or...
  16. V

    Finding Alternative Representations for the Convergent Series Ʃ ((-1)^(i-1))/i

    Homework Statement I basically need to know if the series Ʃ ((-1)^(i-1))/i can be represented in other ways (e.g. a Taylor series, but I doubt it). I know it converges to ln2, but I need to know if there's a series like x^2, x^4, ... or something like it that I can represent the series...
  17. T

    Finding the power series expansion of this ln

    Homework Statement Find the series expansion of ln(x + sqrt(1+x2)) Homework Equations ln(1+x) = x - x2 /2 + x3/3 - x4/4 + ... The Attempt at a Solution I don't know how to solve this. If it was ln(1+f(x) ) I know I could substitute the x's for f(x) in the ln(1+x) series...
  18. J

    Identify power series with coeffs. that are palindromic polynomials of a param.

    Can someone help me to identify the type of power series for which the coefficients are palindromic polynomials of a parameter? More specifically, for a particular function f(x;a) with x, a, and f() in ℝ1, a > 0, and an exponential power series representation F0 + F1x/1! + F2x2/2! +...
  19. C

    MHB Is this power series a convergent series?

    Hi everyone! I have got this series in a part of my research. I need to check if this is a convergent series and if so, what is the radius of the convergence? Here is the series.. \[\sum_{i=0}^{\infty}(-1)^{i}{b-1\choose i}B(y+ac+ci,\,n-y+1)\] Sorry if my LateX code is not visible( I am...
  20. ElijahRockers

    Power Series Solutions of D.E.s

    Homework Statement Find two linearly independent power series solutions about the ordinary point x=0 for y'' + x2y' +y =0 The Attempt at a Solution Alright so we are supposed to try y(x) = Ʃ∞n=0 {Cn(x-x0)n} [but x0=0 so i won't include it in the derivatives] so y'(x) = Ʃ∞n=1...
  21. W

    Solving Initial Value Problem using Power Series Method

    Solve the following initial value problem using a power series representation of the solution around x=0. Find the recurrence relation and the first five nonzero terms of the series solution. d^2y/dx^2 + (2+x) dy/dx +4y=0 ; y(0)=1 ,y'(0)=0
  22. S

    Making a program to convert K(1+cos(nQ-phi)) to a power series

    Hi all ! I am a summer trainee and am currently making (modifying: I am changing the way the program takes input) a program in Fortran, and my math is a little shaky. What I have is an input file with 3 parameters in it for the equation (which is the equation for calculating torsional...
  23. P

    What is the significance of equal coefficients in power series?

    Homework Statement Given \overset{\infty}{\underset{n=0}{\sum}}a_{n}(x-a)^{n} and \overset{\infty}{\underset{n=0}{\sum}}b_{n}(x-a)^{n} that are in R. Then, \overset{\infty}{\underset{n=0}{\sum}}a_{n}(x-a)^{n}=\overset{\infty}{\underset{n=0}{\sum}}b_{n}(x-a)^{n} if and only if a_{n}=b_{n}...
  24. F

    Finding two LI solutions by power series

    Homework Statement y'' - xy' + x²y = 0 Homework Equations y = Ʃ An*x^n (from 0 to infinity) y' = Ʃ n*An*x^n-1 (from 1 to infinity) y'' = Ʃ n*(n-1)*An*x^n-2 (from 2 to infinity) The Attempt at a Solution Ʃ n*(n-1)*An*x^n-2 (from 2 to infinity) - Ʃ n*An*x^n (from 1 to...
  25. S

    Taylor series expansion of a power series.

    If f(x) is a power series on S = (a-r, a+r), we should be able to expand f(x) as a taylor series about any point b within S with radius of convergence min(|b-(a-r)|, |b - (a + r)|) Does anyone have a proof of this or a link to a proof? I have seen it proved using complex analysis, but I...
  26. T

    Find out where this power series converges

    Homework Statement Find out where this power series converges. Ʃ(xn2n) / (3n + n3) Homework Equations The Attempt at a Solution I'm trying to use the ratio test to solve it. I end up with the following equation, which I am unable to reduce further: pn = 2x (3n +...
  27. E

    Solving an Object Falling with Non-Linear Power Series ODEs

    Homework Statement I have not had luck in finding a solution that describes an object falling. Forces include gravitational force which is constant and a vicous force directly proportional to the cube of the velocity. I am supposed to find v as a function of time.Homework Equations v' +...
  28. K

    Use Differentiation to Find a Power Series Representation for:

    Homework Statement for a.) f(x) =1/ ( (1+x)^2 ) what is the radius of convergence? b.) Use part a.) to find a power series for f(x)=1/ ( (1+x)^3) c.) Use part b.) to find a power series for f(x) =x^2 /( (1+x)^3) Homework Equations I want to check my work. I used properties of functions...
  29. A

    Write a partial sum for the power series,

    Write a partial sum for the power series, URGENT Consider the function ln(1+4x). Write a partial sum for the power series which represents this function consisting of the first 5 nonzero terms. For example, if the series were Sigma from n=0 to infinity of 3^nx^2n , you would write...
  30. B

    Exploring the Convergence of a Power Series through Differentiation

    Homework Statement Explicitly compute the function g defined by: g(x) = \Sigman2x2n from n=1 to infinity I was thinking something along the lines of differentiating\Sigma x2n twice
  31. M

    Find the interval of convergence for the given power series.

    Homework Statement Find the interval of convergence for the given power series. Sum from n=1 to infinty of (x-11)^n / (n(-9)^n) Homework Equations The Attempt at a Solution I used the ratio test and I'm getting 2<x<20, but that doesn't seem to be right. I get abs(1/9*(x-11)) <...
  32. M

    Power series involving arctan(x)

    Homework Statement The function f(x) = 8x*arctan(6x) is represented as a power series f(x) = sum from n=0 to infinity of Cn * x^n Find the first few coefficients in the power series Homework Equations The Attempt at a Solution I deduced that 8xarctan(6x) = sum from n=0 to...
  33. R

    Representing Airy's function as a power series

    Homework Statement Find the first five non-zero terms of the power series solution to d2y/dx2-xy=0 about x=-2; y(-2)=1;y'(-2)=1/2 Homework Equations ... calculus in general? and the taylor expansion of y(x) is - assuming remainder term is zero: \sumy(n)(-2)/n! *(x+2)n (n from 0 to...
  34. D

    MHB How to Prove the Sum of the Power Series Using the Series for Cotangent?

    Show $\sum\limits_{n=1}^{\infty}\frac{1}{n^2} = \frac{\pi^2}{6}$ using the series for $(\pi\cot\pi z)'$ at $z = 0$ I know from class that $\sin\pi z = \pi z\prod\limits_{n\in\mathbb{Z} -\{0\}}\left[\left(1-\frac{z}{n}\right)e^{z/n}\right]$ So do I need to use that to rewrite cot as cosine over...
  35. S

    Theoretical/non-tedious question about power series solution of y'' + y = 0

    1. "Homework Statement Find a recurrence formula for the power series solution around x = 0 for the differential equation given in the previous problem." The previous problem says: "Determine whether x = 0 is an ordinary point of the differential equation y'' + y = 0." Homework...
  36. A

    Relation of laplace transform with power series

    Hi, just wonder if anyone can help Homework Statement Apparently there is a relation between laplace transform and power series. http://www.jstor.org/stable/pdfplus/2305640.pdf?acceptTC=true states that if the discrete variable n of a power series is replaced by a continuous variable lambda...
  37. T

    Comp Sci Power series with subprograms Fortran 77 HELP

    Homework Statement The power series 1 + x + (X^2)/2! + (x^3)/3! +...(to infinity)= (x^k)/k! converges to e^x for all values of x. Write a function subprogram that uses this series to calculate values for e^x to five-decimal-place accuracy (i.e. using terms up to the first one that is less...
  38. S

    How do you find the derivatives of a power series?

    Homework Statement http://imgur.com/FJhgN Give this power series J(x) (leftmost in the picture), find the first and second derivative. Homework Equations You take the derivative of a power series term by term. The Attempt at a Solution I don't understand how to get the J''(x) in the...
  39. L

    Radius of Convergence of power series anx^n^2

    Homework Statement Suppose that the power series \sumanxn for n=0 to n=∞ has a radius of convergence R\in(0,∞). Find the radii of convergence of the series \sumanxn2 from n=0 to n=∞ and \sumanx2n.Homework Equations Radius of convergence theorem: R = 1/limsup|an|1/n is the radius of...
  40. R

    Power series representing ∫sinx/x

    Homework Statement Find the Power Series representing g(x)=∫sin(x)/x Homework Equations sin(x)= x-(x^3/3!)+(x^5/5!)-(x^7/7!) The Attempt at a Solution I Havent attempted yet but was wondering if you start with the maclaurin series of sin(x) then divide everything by x then...
  41. C

    Power Series of arcsin: Finding Radius and Interval of Convergence

    Homework Statement a. Find a power series expansion for arcsin(x) centered at 0. b. Find the radius of convergence and interval of convergence of the power series in a. c. Choose an appropriate value of x to plug into the power series found in a. to find a series that converges to...
  42. H

    Deriving the Power Series of Arctan through Integration

    How do we get the power series of arctan using integration. Could someone explain this step by step as I'm quite confused about it.
  43. L

    Differential Equation Power Series Method

    Differential Equation Power Series Method (Almost done!) Homework Statement solve (2x-1)y'+2y=0 using power series. I'm really close to the correct answer, which is c/(1-2x). I keep getting 2c/(1-2x). I got the correct radius of convergence, however (1/2) Homework Equations shown in my...
  44. M

    What Does the Subscript in the Airy Function Power Series Notation Mean?

    I am currently working on a solution to an differential equation of the form I(x)-xI(x)=0. The solution is the airyai and airybi functions, and I have found the power series equations for these. I am using two different mathematical programs to evaluate the solution, and each are giving me...
  45. M

    Mass-spring-damper system solve for x(t) using power series

    Homework Statement A mass of 10kg is suspended from vertical spring of stiffens 100N/m and is provided with dashpot damper having damping coefficient of 1000Ns/m. The mass is pulled down the distance of 4cm from its equilibrium position and than released. Establish the differential equation...
  46. L

    Understanding the Identity Theorem for Power Series Coefficients

    Hey guys, I've been trying to work out this question, http://img189.imageshack.us/img189/2954/asdagp.jpg so the identity theorm is just that if the power series = 0 then the coefficient of the series must be zero. Im having trouble seeing how that negative has any influence over...
  47. Telemachus

    Nonhomogeneous Power Series Solution

    Hi. I have to solve: y''+xy'-2y=e^x Using series. So, this is what I did: y(x)=\sum_0^{\infty}a_n x^n y'(x)=\sum_1^{\infty}n a_n x^{n-1} y''(x)=\sum_2^{\infty}n(n-1) a_n x^{n-1} And e^x=\sum_0^{\infty}\frac{x^n}{n!} Then, using that m=n-2 for y'' and then replacing in the diff. eq...
  48. D

    MHB What is the correct power series for $g(t)$?

    What would be the power series of $g(t)$? $$ g(t) = \sum_{n=0}^{\infty}\frac{g(t)}{n!} $$ This?
  49. S

    Finding a power series expansion for a definite integral

    Homework Statement Find a power series expansion about x = 0 for the function f(x) = ^{1}_{0}\int\frac{1 - e^{-sx}}{s} ds Homework Equations The power series expansion for a function comes of the form f(x) = ^{\infty}_{0}\sum a_{k}x^{k} The Attempt at a Solution I've tried...
  50. S

    Finding general solution to a differential equation using power series

    Homework Statement Find the general solution near x = 0 of y'' - xy' + 2y = 0 (using power series). Answer: y = a_0 * y_1(x) + a_1 * y_2(x) where y_1(x) = 1 - x^2 and y_2(x) = x - 1/6 * x^3 - 1/120 x^5 - 1/1680 * x^7 - ... Homework Equations Power series. Sigma notation for...
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