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

    Finding a Power Series for a function

    Homework Statement Find a power series representation for the function f(x) = \frac{(x-1)}{(3-x)^2}^2, valid for every x with |x|<3Homework Equations The equation that I think would be useful is \frac{1}{1-x} = \sum_{n=0}^\infty x^n The Attempt at a Solution I began by just looking at the...
  2. M

    Conversion of a function to a power series

    Given the function f(x) = 1/(3-x) it can be represented in a power series by 1/3∑(x/3)n but is there any restriction on saying ∑(x-2)n except for x = 2? In the first case, R = 3 but the second case, R = 1 and on different intervals (i.e. (-3,3) and the other is (1,3). I just simply used the...
  3. vanceEE

    How Do You Solve the Differential Equation y' = xy Using Power Series?

    Homework Statement $$y' = xy$$Homework Equations $$y = a_{0} + a_{1}x + a_{2}x^{2}+... = \sum\limits_{n=0}^∞ a_{n}x^{n}$$ $$xy = a_{0}x + a_{1}x^{2} + a_{2}x^{3}+... = \sum\limits_{n=0}^∞ a_{n}x^{n+1}$$ $$y' = a_{1} + 2a_{2}x + 3a_{3}x^{2}... = \sum\limits_{n=1}^∞ n a_{n}x^{n-1}$$The Attempt...
  4. A

    Power series, formal power series and asymptotic series

    What's the difference between these three? I only know Taylor series and its variants which I suppose is called power series (but I'm not sure). In that you just approximate around a single point using derivatives. But what are formal powers series and asymptotic expansion? I did see...
  5. A

    Quick question on power series of secant

    Hey, everyone. I am trying to find the power series of secant from the known power series of cosin, but my answer does not match up with Wolfram and Wikipedia. I know: cos(\theta) = 1 - \frac{1}{2}x^2 + \frac{1}{4!}x^4 + ... So, using the first two terms (assuming a small angle)...
  6. A

    Finding the power series of a square root

    Homework Statement Find a power series for f(x) = \frac{1}{\sqrt{4+x^{2}}}, at x=0. 2. The attempt at a solution I have looked up the Taylor series of \frac{1}{\sqrt{4+x^{2}}}, but I don't find any similarity with a power serie like \sum_{n\geq 0} a_{n} x^{n} I don't know how to start...
  7. J

    Power series for integral (1/x) dx

    Homework Statement I have to find the power series representation for integral (1/x) dx Homework Equations ln (1+x) The Attempt at a Solution This is very similar to ln(1+x) but I don't know if this helps me. Is this ln(x) shifted one to the right? So maybe I can use what is...
  8. J

    Power Series Question | Limit and Convergence | Solution Attempt

    Homework Statement Question. Did I do this OK? Homework Equations The Attempt at a Solution A_n = Ʃ e^(n^2) x^n from n = 1 to ∞ So I tried the root test. After you take the nth root you have x e^n so then I took the limit of this as n-->∞ and I got infinity. I then said OK...
  9. C

    Power Series Solution to Linear ODE

    Homework Statement Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of (x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0 Note that x=0 is an ordinary point. Homework Equations y(x)=\sumckxk (k=0 to ∞) y'(x)=\sum(kckxk-1) (k=1 to ∞) y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞) The Attempt at a Solution...
  10. R

    Finding Recursion Relations for Coefficients in Power Series Solutions for ODEs

    Homework Statement I am trying to find the recursion relation for the coefficients of the series around x=0 for the ODE: y'''+x^2y'+xy=0 The Attempt at a Solution Therefore letting: y=\sum_{m=0}^\infty y_mx^m \therefore y'=\sum_{m=1}^\infty my_mx^{m-1} \therefore...
  11. G

    Can the constant term of a power series be zero?

    In the context of my work (linear differential equations), it can not be zero. But why?
  12. C

    Second Order Linear ODE - Power Series Solution to IVP

    Homework Statement Let y(x)=\sumckxk (k=0 to ∞) be a power series solution of (x2-1)y''+x3y'+y=2x, y(0)=1, y'(0)=0 Note that x=0 is an ordinary point. Homework Equations y(x)=\sumckxk (k=0 to ∞) y'(x)=\sum(kckxk-1) (k=1 to ∞) y''(x)=\sum(k(k-1))ckxk-2 (k=2 to ∞) The Attempt at a Solution...
  13. Feodalherren

    General question about differentiating power series

    Say I have a simple series like \Sigma^{∞}_{n=0} X^{n} When I differentiate this series the first term goes to 0 because it's a constant. Does that mean that I have to adjust the index of the series from n=0 to n=1? If I don't do it, the first term still goes to zero as n(x^(n-1)) when n=0...
  14. N

    Radius of convergance complex power series

    Homework Statement https://scontent-a.xx.fbcdn.net/hphotos-ash3/1390611_10201748262844961_2141774184_n.jpg I need help with 7b. Theorem 3 = termwise differentiation and theorem 4 = termwise integration.Homework Equations The Attempt at a Solution I have no idea how differentiation or...
  15. G

    How Can I Find the Power Series Representation of the Given Integral Function?

    Hi, I'm trying to find the series representation of f(x)=\int_{0}^{x} \frac{e^{t}}{1+t}dt . I have found it ussing the Maclaurin series, differenciating multiple times and finding a pattern. But I think it must be an eassier way, using the power series of elementary functions. I know that...
  16. V

    2nd Order Differential Equation via Power Series

    Homework Statement Problem 5 on the attached Sheet here Homework Equations We studied the Power Series Method and how to calculate a linearly independent solution if one solution is already known. So we need to find one solution (probably) using the power series method and then using...
  17. Y

    Finding the Closed Form of a Power Series

    Homework Statement Using that \frac{1}{1-x} = \sum_{n=0}^{\infty} x^n for |x|<1 and that f'(x) =\sum_{n=0}^{\infty} (n+1)a_{n+1}(x-x_0)^n , write \sum_{n=0}^{\infty} n^2x^n in closed form. Homework Equations The Attempt at a Solution In this series, a_n = n^2 and x_0 = 0 ...
  18. N

    MHB Expanding to power series, and finding the Laurent Series

    Please refer to attached image. Hi, I'm a bit lost here with the first question. Unfortunately the online lecture covering this material isn't available due to their having been made some technical difficulties, and I find our textbook difficult to comprehend! My lecture notes are pretty...
  19. B

    Non-linear convolution and power series

    Homework Statement Hi, suppose we have the summation \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} a_j b_{i-j}^{j} x^i, where the subscripts are taken modulo n, and a_i^n = a_i, b_i^n = b_i for each i. Write the above power series as a product of two power series modulo x^n - x.Homework Equations I'm...
  20. J

    Antiderive complex function f(z) and express as power series

    Let F(z) be the anti-derivative of the function f(z) = cos(z^3) with F(0) = 0. Express F(z) as a power series around z=0, giving both the first 3 non-zero terms and the general (nth) term. Hey guys really struggling with this integration and how to then express this as a power series. Any...
  21. J

    Expressing a function as a power series

    Hey guys! Suppose you have a function f(x)=1/2-x which you need to express as a power series. I am familiar with the conventional way of solving its series form, which involves taking out 1/2 from f(x) and arriving with a rational function 1/1-(x/2) which is easy to express as a power series...
  22. J

    26th Derivative of a Function- Power Series

    Homework Statement I cannot write out the equation clearly so I am attaching a file. Homework Equations The Attempt at a Solution sin x= x- x^3/3! + x^5/5! - x^7/7! + ... sinx/(x)= 1- x^2/3! + x^4/5! - x^6/7! - x^26/27! + ... (-1)^k x^(2k) / (2k+1)! = g^(2k)(0) x^(2k)/(2k)...
  23. S

    Differential Equations - Power Series problem with e^t

    Homework Statement The problem is to solve: y''+ty'+e^{t}y=0, y(0)=0 and y'(0)=-1 Homework Equations The Attempt at a Solution My main issue is the following: I normally find the recursion relation, and then factor out the t^{whatever} and I know that the coefficient to this...
  24. T

    How Do You Find the Power Series Expansion of \( e^z \) at \( \pi i \)?

    "[F]ind the power-series expansion about the given point for each of the functions; find the largest disc in which the series is valid. 10. ##e^{z}## about ##z_{o} = \pi i##" (Complex Variables, 2nd edition; Stephen D. Fisher, pg. 133)$$f(z) = e^{z} = e^{z-a} \cdot e^{a} = e^{a} \cdot \sum...
  25. M

    Understanding Parenthesis Style in Power Series Questions

    Hello, My question is about power series. In most of questions i can find points with ratio test. But when i check points i can't understand style of parenthesis. Is there easy way? For example:\sum(n^(3)*(x-5)^n) I found check points 4<x<6 How can i decide to the parenthesis will be...
  26. A

    Calculating Improper Integral w/ Power Series of r=1

    At exam today I was to calculate an improper integral of a function f defined by a power series. The power series had radius of convergence r=1. Inside this radius you could of course integrate each term, i.e. symbologically: ∫Ʃ = Ʃ∫ The only problem is that the improper integral went from 0...
  27. A

    Does Using Maximum Coefficients Determine the Smallest Radius of Convergence?

    Homework Statement Let Ʃanx^n and Ʃbnx^n be two power series and let A and B be their converging radii. define dn=max(lanl,lcnl) and consider the series Ʃdnx^n. Show that the convergence radius of this series D, is D=min(A,B) Homework Equations My idea is to use that the series...
  28. vibhuav

    Power series summation equation

    (Was posted in general physics forum also) I am currently reading Roger Penrose’s “Road to Reality”. In section 4.3, Convergence of power series, he refers to the sum of the series: 1 + x2 + x4 + x6 + x8 + ... = 1/(1-x2) Of course, this is true for |x| < 1, beyond which the series...
  29. A

    Radius of convergence of power series

    Homework Statement The coefficients of the power series \sum_{n=0}^{∞}a_{n}(x-2)^{n} satisfy a_{0} = 5 and a_{n} = (\frac{2n+1}{3n-1})a_{n-1} for all n ≥ 1 . The radius of convergence of the series is: (a) 0 (b) \frac{2}{3} (c) \frac{3}{2} (d) 2 (e) infinite Homework EquationsThe Attempt at...
  30. W

    How can the Power Series for Arc Tan be Proven for Homework?

    Homework Statement Prove. Homework Equations arctan(x) = x - \displaystyle \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \frac{x^9}{9} for -1 < x < 1. The Attempt at a Solution \displaystyle \sum^{∞}_{n=1} \frac{x^{n+2}}{n+2}
  31. M

    Asymptotic behavior of a power series near its branch point

    I was reading a paper the other day that made the following claim, and provided no reference for the assertion. I would like to find a reference or figure out how to derive the asymptotic behavior myself. The situation is as follows: Suppose we have a function ##f(z)##, defined as a power...
  32. M

    Finding the Power Series of f'(x) from f(x) = x^2cos(2x)

    Homework Statement Find the power series of f'(x), given f(x) = x2cos2(x) Homework Equations Correct me if I'm wrong The Attempt at a Solution Can I just take the derivative of the solution I got previously? If so, what's a good way to write the sequence out so I can easily...
  33. I

    Power Series Convergence Question

    Homework Statement Can anyone explain to me why the answer to this question is D?: http://puu.sh/2FoET.png The Attempt at a Solution I'm not really sure where to begin, except I know that the series is centered at 0. I was also thinking that the given x's was the Interval of...
  34. D

    Power Series Representation of (1+x)/(1-x)

    Homework Statement For the power series representation of, f(x)=1+x1−x which is 1+2∑from n=1 to inf (x^n), Where does the added 1 in front come from? How do I get to this answer from ∑n=0 to inf (x^n)+∑n=0 to inf (x^(n+1)) Homework Equations The Attempt at a Solution I arrived at ∑n=0 to inf...
  35. W

    Power Series Converge Absolutely

    Homework Statement for what values of x does the series converge absolutely?Homework Equations \displaystyle \sum^{∞}_{n=1} \frac{4^n * x^n}{n!}The Attempt at a Solution Ratio Test \displaystyle \frac{4^{n+1} * x^{n+1}}{n+1)!} * \frac{n!}{4^n * x^n} 4x * limit (n->inf) \displaystyle...
  36. X

    Radius of Convergence Power Series

    Homework Statement Determine the radius of convergence and the interval of convergence og the folling power series: n=0 to infinity Ʃ=\frac{(2x-3)^{n}}{ln(2n+3)} Homework Equations Ratio Test The Attempt at a Solution Well I started with the ratio test but I have no clue where...
  37. Fernando Revilla

    MHB Champ's question at Yahoo Answers (Power series)

    Here is the question: Here is a link to the question: Find a power series... Calc Help? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  38. STEMucator

    Convergence of Arctan(t) Power Series at Endpoints

    Homework Statement An exercise from advanced calculus by taylor : Homework Equations The Attempt at a Solution (a) ##\int_{0}^{x} tan^{-1}(t) dt = \int_{0}^{x} \sum_{n=0}^{∞} (-1)^n \frac{t^{2n+1}}{2n+1} dt = \sum_{n=0}^{∞} \frac{(-1)^n}{2n+1} \int_{0}^{x} t^{2n+1} dt =...
  39. K

    A question regarding the Power series method of solving the QHO

    I am working through the Griffiths QM text and I am getting caught up on some the process he uses to derive the wave functions and energy levels for the QHO, via Frobenius/Power series method. I understand that the Schrodinger equation get recast into a summation form over the coefficients...
  40. S

    Solving an ODE with power series method

    Homework Statement Solve ##(1-x)y''+y=0## at the point ##x_0=0##. Use this solution to find a solution to ##xy''+y=0## around the point ##x_0=1##. Homework Equations The Attempt at a Solution ##(1-x)y''+y=0## ##(x-1)y''=y## ##\displaystyle\sum_{k=2}^\infty a_k k...
  41. Z

    Power Series from MTW 19.1: Explained?

    hello Please see attachment which is a snapshot from MTW first page of chapter 19. Can someone please elaborate on how the equations 19.3b and c can be explained ? I know that equation 19.3a is a familiar formula but not so much the other two. I'm just confused. Thank you, Long live & clear...
  42. I

    Power Series - Finding x values for which the series equals a certain number

    Homework Statement Given x-\frac{x^2}{2}+\frac{x^3}{3}-\frac{x^4}{4}+... = k, are there any values of x for the values k = -100, \frac{1}{2}, 100? Homework Equations The Attempt at a Solution I started by finding that the series is \Sigma^{\infty}_{n=1} \frac{(-1)^{n+1}x^n}{n} = k...
  43. S

    Power Series Representation for x/(15x^2+1): Is My Solution Correct?

    Homework Statement write a power series representation of the following: \frac{x}{15x^2 +1} Homework Equations the formula \frac{1}{1-x} = 1 + x + x^2 + ... = \sum_{n=0}^{∞} x^n The Attempt at a Solution we can rewrite the summnd like \frac{x}{15} \left(...
  44. M

    Frobeniuns Method/Generalized Power Series to DiffEQ solutions

    (Working out of Boas chapter 12, section 11) 3xy'' + (3x + 1)y' + y = 0 I'm asked to solve the differential equation using the method of Frobenius but I'm finding the way Boas introduces/explains/exemplifies the method to be incredibly confusing. So, I used some google-fu and was even...
  45. R

    Showing Relation of e^(ipa/ħ)xe^(-ipa/ħ)=x+a Using Power Series

    1. The problem statement: Show that if the operator relation e^(ipa/ħ)xe^(-ipa/ħ) = x+a holds. The operator e^A is defined to the ∞ e^A= Ʃ(A^n)/n! n=0 [Hint: Calculate e^(ipa/ħ)xe^(-ipa/ħ)f(p) where f(p)is any function of p, and use the representation x=iħd/dp]...
  46. T

    How Do You Simplify and Find the Interval of Convergence for a Power Series?

    Edit: Nevermind, figured it out. Thank you for readingOriginal problem: Find the interval of convergence \sum∞n=1 xn / n * √(n) * 3n Ratio Test, right? an+1/a I get to here and I can't figure out how to get rid of the ns: lim n→∞ abs(x/3)* [n*√(n) / (n+1)*√(n+1)] Solution, They break apart...
  47. S

    MHB Power series solution for Log(1+x)

    Show that, \[\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}+\cdots\]
  48. P

    Finding the Sum of a Power Series: Tips and Tricks for Success

    Homework Statement I am trying to find the sum of the series in the attachment. Homework Equations The Attempt at a Solution I have tried to use various series and their derivatives, to not much avail. I am not sure how to handle the n^2 factor. Should I break it down to two...
  49. P

    Proving Convergence of Power Series for All x within Radius of Convergence

    Hi, Homework Statement I am asked to prove that if the power series Ʃ(1,infinity) a_n(x-x0)^n converges at a point d, then it converges for every x that satisfies |x-x0|<|d-x0|. Homework Equations The Attempt at a Solution Obviously |d-x0|<r, where r denotes the radius of...
  50. M

    Power Series Identity for Bessel Functions

    Homework Statement Show e^{\frac{x}{2}(t-\frac{1}{t})}=\sum^{\infty}_{n=-\infty}J_n(x)t^n Homework Equations J_k(x)=\sum^{\infty}_{n=0}\frac{(-1)^n}{(n+k)!n!}(\frac{x}{2})^{2n+k} The Attempt at a Solution Power series product (\sum^{\infty}_{n=0}a_n)\cdot (\sum^{\infty}_{n=0}...
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