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. P

    Does Boundedness Affect the Radius of Convergence for Subtracted Power Series?

    Say we have two power series \sum_{n=0}^{\infty}a_n z^n and \sum_{n=0}^{\infty}b_n z^n which both converge in the open unit disk. Is there anything we can say about the radius of convergence of the power series formed by their difference? i.e. \sum_{n=0}^{\infty}(a_n-b_n) z^n What about if we...
  2. Char. Limit

    Power Series Solution for y' = 4xy + 2 with Initial Condition y(0)=1

    Homework Statement I am trying to find the power series solution to y' = 4 x y + 2, with the initial condition of y(0)=1. Homework Equations The Attempt at a Solution Simple enough, I say, as I arrange the equation so I have 0 on one side. I get something like this: y' - 4 x y - 2 = 0 I then...
  3. S

    Find a recurrence formula for the power series solution around t = 0

    Homework Statement Find a recurrence formula for the power series solution around t = 0 for the differential equation: d^2 y/dt^2 + (t - 1) dy/dt + (2t - 3)y = 0 Homework Equations y = Σn=0 to inf (a_n * t^n) and formula to differentiate polynomials. The Attempt at a Solution I...
  4. F

    Finding the Power Series for z^7

    How do I find the power series for z^7? I can't remember.
  5. alexmahone

    MHB Convergence Condition for Applying Ratio Test to Power Series

    Given a power series, what is the condition on its coefficients that means the ratio test can be applied?
  6. A

    Find a power series for the function

    Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x) = arctan(x/3) Homework Equations not really any equations ... just intervals The Attempt at a Solution here's what i did: f'(x) = \frac{1}{1+(x/3)^2} =...
  7. A

    Calc 2: Power Series - Find Radius & Interval of Convergence

    Homework Statement Find the radius of convergence and interval of convergence of the series: \sum (-1)n * ( x^(2n) / (2n)! ) for n=0 to infinity Homework Equations well, any of the divergence tests: ratio, limit comparison, etc. The Attempt at a Solution I don't get power...
  8. P

    Proof of commutative property in exponential matrices using power series

    I'm trying to prove eA eB = eA + B using the power series expansion eXt = \sum_{n=0}^{\infty}Xntn/n! and so eA eB = \sum_{n=0}^{\infty}An/n! \sum_{n=0}^{\infty}Bn/n! I think the binomial theorem is the way to go: (x + y)n = \displaystyle \binom{n}{k} xn - k yk = \displaystyle...
  9. P

    MHB Why is the Maclaurin series of $e^u$ simply $\sum\frac{u^n}{n!}$?

    It just occurred to me now that I never asked why the Maclaurin series of $e^u$, u being some function of x, is simply $\sum\frac{u^n}{n!}$. I should say I understand why the Maclaurin series of $e^x$ is $\sum\frac{x^n}{n!}$, due to the derivative of the exponential function being itself, but...
  10. J

    Complex Power Series Radius of Convergence Proof

    Homework Statement If f(z) = \sum an(z-z0)n has radius of convergence R > 0 and if f(z) = 0 for all z, |z - z0| < r ≤ R, show that a0 = a1 = ... = 0. Homework Equations The Attempt at a Solution I know it is a power series and because R is positive I know it converges. And if...
  11. S

    MHB Power Series Expansion: Solving for 1/1+z at z=-5 with Radius of Convergence

    I'm trying to find the power series expansion of 1/1+z at z=-5 and the radius of convergence.How should I think and solve this problem? I'm looking for a step by step explanation because I want to understand the mechanics behind it.Thank you.
  12. D

    MHB How do I expand $f(z)$ into a power series?

    $\displaystyle f(z) = \frac{4 + 3z}{(z + 1)(z + 2)^2}$ How do I find the power series? I know that $\displaystyle\frac{1}{z+1} = \frac{1}{1-(-z)} = \sum_{n}^{\infty}(-z)^n$ and $\displaystyle\frac{1}{(z+2)^2} = \frac{d}{dz} \frac{-1}{z+2} = \frac{d}{dz} \frac{-1}{1 - (-z-1)} =...
  13. F

    Power Series for Complex Number z: f(z) and Radius of Convergence Calculation

    z is a complex number. f(z) = \frac{4 + 3z}{(z + 1)(z + 2)^2} \frac{1}{1 + z} = \frac{1}{1 - (-z)} = 1 + (-z) + (-z)^2 + \cdots = \sum_{n = 0}^{\infty}(-z)^n \frac{1}{(z + 2)^2} = -\frac{d}{dz} \ \frac{1}{1 - (-z - 1)} = -\frac{d}{dz}\sum_{n = 0}^{\infty}(-z - 1)^n = \sum_{n =...
  14. F

    Exploring the Power Series of $\frac{1}{1-(-z)}$

    z\in\mathbb{Z} \frac{1}{1-(-z)}=\sum_{n=0}^{\infty}(-z)^n \frac{1}{(z+2)^2}=\frac{d}{dz} \frac{-1}{1-(1-z)} = \frac{d}{dz} (1 + (1-z) + (1-z)^2+\cdots = 0 -1 -2(1-z)-3(1-z)^2 - \cdots = \sum_{n=0}^{\infty} ? Not to sure about the second one.
  15. C

    Hard power series and initial value problem question

    Homework Statement We know that y = Aex is the solution to the initial value problem dy/dx = y; y(0) = A. This can be shown by solving the equation directly. The goal of this problem is to reach the same conclusion using power series. Method: Let y be a solution to the initial value...
  16. N

    Non-convergent power series but good approximation?

    Hello, In my QM class we're using power series which don't converge but apparently still give a good approximation if one only takes the lower-order terms. Is there any way to understand such a phenomenon? Is it a genuine area of mathematics? Or is it impossible to say something general on...
  17. J

    Solving Power Series with Integral Test for Homework

    Homework Statement My professor says that the integral test is supposed to help me figure out the interval of convergence of this equation: Ʃ(2n*(x+1)n)/(n*ln(n)) Homework Equations integral test The Attempt at a Solution i couldn't figure out what the power series was after i...
  18. J

    Question about radius of convergence of fractional power series

    Suppose I have the Laurent series with region of convergence given below: f(z)=\sum_{n=-\infty}^{\infty} a_n z^n,\quad \sqrt{3}<|z|<\sqrt{5} Can I conclude the Laurent-Puiseux series: f(\sqrt{z})=\sum_{n=-\infty}^{\infty} a_n \left(\sqrt{z}\right)^n has a region of convergence...
  19. C

    Expanding Power Series for (x+x^2)/(1-x)^3 | Simplification Techniques

    Homework Statement Expand f(x)= (x+x2)/(1-x)3 Homework Equations ? The Attempt at a Solution I've tried everything I can think of to simplify this equation: substitution of various other power series, partial fraction decomposition, taking derivatives, multiplying out the...
  20. P

    Find Sum of Power Series: Hint and Tips

    I have to find the sum of the power series: \sum_{n=1}^\infty nx^{n+1} I know the I'm supposed to show work but I don't have any idea where to start. I'm not asking for you to do the problem for me, just a hint. The only idea I had was to take the derivative to get rid of the n+1 in the...
  21. P

    Power Series Representation of xln(1+x^2)

    Write a power series representation of xln(1+x2) My first instinct was to attempt to take the second derivative and then find the summation and then integrate but that approach seemed to be a dead end. Basically, the x thrown in there confuses me and you can't split the function into two...
  22. J

    How to Solve for n in a Power Series Limit?

    Power Series:(n^n)*(x^n) Homework Statement The only step I'm having problem with on this problem is when I take the Lim n→∞ of the problem. I want to know how to cancel the n^n on the denominator during one of the steps. Homework Equations ∞ Ʃ (n^n)*(x^n) n=1 The Attempt at a...
  23. G

    Prove the following Power Series is monotonic

    f(x) = \sum_{n=1}^{\infty} \frac{(-1)^nx^{n}}{(n)^{\frac{3}{2}}} Prove f(x) is strictly monotonic (where f is defined) and that there exists one solution to f(x)=1.5 and f(x)=-0.5 First, how do I show that the radius of convergence is between -1 \le x \le 1? And then, how do I...
  24. P

    Analysis: Absolute convergence, rearranging terms, power series question

    Homework Statement I can't figure out what my professor means in the last two lines: I'm trying to prove why the product of two analytic functions is analytic, and I think that I am going to need to use a similar construction. The Attempt at a Solution So far to prove the product...
  25. K

    Power Series x=0: What Does a0 & x0 Mean?

    what does it mean if y(x)=\sum^{∞}_{n=0} an(x-x0)n and y(0)=1 ? does this mean that x0=0 and a0=1 ?
  26. K

    Solve differential equation using power series

    Find y(x) as a power series satisfying: y'-2xy=0 , y(0) = 1 attempt: y=\sum^{∞}_{n=0} an(x-x0)n y'=\sum^{∞}_{n=0} (n+1)an+1(x-x0)n y' - 2xy = y' - 2y(x-x0) - 2x0y substituting the power series into above formula and simplifying eventually gives: a1-2x0a0 +...
  27. C

    Power Series Expanded, Arfken 5.7.16

    Homework Statement The behavior of a neutron losing energy by colliding elastically with nuclei of mass A is described by a parameter ξ1, ξ1 = 1 + \frac{(A-1)^2}{2A}*ln\frac{A-1}{A+1} An approximation, good for large A, is ξ2= \frac{2}{A+2/3} Expand ξ1 and ξ2 in powers of A−1. Show...
  28. T

    Complex Power Series Convergence Help

    Homework Statement I have a problem set that asks me to determine, first, the radius of convergence of a complex series (using the limit of the coefficients), and second, whether or not the series converges anywhere on the radius of convergence. Homework Equations As an example: Σ(z+3)k2 with...
  29. F

    Heat Capacity Power Series Approximation

    Homework Statement "Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x^{3} in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than...
  30. E

    Factorial question in a power series solution

    Hello, I've been working on solving the equation y''-2xy'+2py=0. where p is a positive integer. I've assumed y=\sum a_{n}x^{n} for n=0 to inf I'm getting two formulas for a_{n} One is for odd n, the other for even n, related to a_{0} and a_{1} However, the relation involves something that...
  31. K

    Radius & interval of convergence of power series

    Homework Statement i was doing this exercise and came across this example. ∞ Ʃ (x^n)/ln(n+1) n=1 The Attempt at a Solution i know you have to do the ratio test which is lim | a(n+1)/a(n)| n>∞ i got to lim | [x ln(n+1)] /ln(n+2) | n>∞ and have no idea how to continue? is...
  32. B

    Is My Power Series Approach Correct for Solving This Differential Equation?

    I've just started learning to solve DE by using power series and I am not sure If I did it the right way. Would anybody be kind enough going thru my solution here and see if I did it right? I want to make sure I am doing the right thing, I am very bad with series. Thanks in advance!and i have...
  33. I

    Radius of convergence (Power series)

    Homework Statement Hi there, I have just started taylor series for my course.. most seems arlgiht so far, however when it comes to validating a given series( tayor or maclaruin), I have an idea on how to find out the x value.. but I don't know what I am doing wrong.Take for example: The...
  34. T

    Power Series Expansion of Dedekind Eta Function: How to Expand η(τ)/η(3τ)?

    Hi Every body! I wan to compute the power series expansion of dedekind eta function. Specifically, I want to know the power series expansion of η(τ)/η(3τ)? How could I expand this function? I would be happy if you could help me as I am stuck at this state when I am computing the modular...
  35. B

    Determine Series Convergence Given Convergence of a Power Series

    Homework Statement I am asked to comment on the convergence/divergence of three series based on some given information about a power series: \sum_{n=0}^{\infty}c_nx^n converges at x=-4 and diverges x=6. I won't ask for help on all of the series, so here's the first one...
  36. C

    Finding Power Series Representation for f(x) and Interval of Convergence

    Homework Statement Find the power series representation for the function f(x)=x/(x^2-3x+2) and determine the interval of convergence. Homework Equations The Attempt at a Solution First I separate into partial fractions 2/(x-2) - 1/(x-1) 2/(x-2) = sum n=0 to infinity (x/2)^n...
  37. B

    Converting complex power series into a function

    Hey guys, sorry for sending out so many questions so fast. I just discovered this site, and it looks great. Plus, I have my first complex analysis midterm tomorrow, so I'm pretty stressed (you'd think after 4 years of math/econ/computer science you'd get used to it but there's nothing like the...
  38. L

    Exponential Power Series Expansion

    I want to show that e^x e^x = e^{2x} using a power series expansion. So I start with \sum_{n=0}^\infty \frac{x^n}{n!} \sum_{m=0}^\infty \frac{x^m}{m!} \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^n}{n!} \frac{x^m}{m!} \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^{m+n}}{m!n!} I am...
  39. D

    Computing Finite Power Series Sums in MATLAB: A Beginner's Guide

    I having trouble trying to compute the sum of a finite power series with matlab.Basically, the problem says to take x=0.5 and compute the sum of a series for n=5, n=10, n=25, and n=100 and for all coefficients ai=1. It also says to use the operation .^ and command sum(v) that takes a vector and...
  40. J

    Another power series DE question

    Homework Statement Using power series, find the solution to the following DE y" + xy' + y=0 Given data xo = 0 y(0)=1 and y'(0) = 1 Homework Equations y(x) = \suman(x-xo)^n for n= 0 to \infty The Attempt at a Solution Please see attached file. The part circled at the...
  41. J

    Power series differential equations question

    Homework Statement Using power series, find solutions to the following DE y' + y= x^2, y(1)= 2 and xo=1 Homework Equations y(x)=an\sum(x-xo)^n for n=1 to infinity The Attempt at a Solution See the attachment NOTE: I only want to find a way to collect all the x...
  42. G

    What is the radius and interval of convergence for the given power series?

    Homework Statement Find the radius of convergence and the interval of convergence of the series sigma[n=1,inf] ((3x-2)^n/(n^2*3^n)) Homework Equations The Attempt at a Solution sigma[n=1,inf] ((3x-2)^n/(n^2*3^n)) I applied the Root Test p=lim n->inf |(3x-2)^n/(n^2*3^n)|^(1/n) = lim n->inf...
  43. S

    Power series in real world situation

    Homework Statement A heavy weight is suspended by a cable and pulled to one side by a force F. How much force is required to hold the weight in equilibrium at a given distance x to one side. Tcosθ=W and Tsinθ=F. Find F/W as a power series of θ. Often in a problem like this, what we know is not...
  44. S

    Power series when variable is very large

    Homework Statement Find first three non zero terms in series expansion for ln(1+e^-z) when z is very large Homework Equations The Attempt at a Solution I've got as far as ln(1+e^(-1/z)*e^((1/z)(z^(2) - 1)) not sure where to go from here
  45. M

    Find a function for this power series.

    Homework Statement Perform a partial fractions expansion of 1/(n(n+1))=a/n + b/(n+1) in order to find a function that represents x^n/(n(n+1)). Homework Equations The Attempt at a Solution so i broke up the partial fraction to 1/n -1(1+n). I integrate both to get ln(n) +...
  46. P

    Differential equation with power series help

    Homework Statement (1+2x^2)y''+6xy'+2y=0 1. find the power series solutions of the equation near x0=0...show the recurrence relation for an, derive a formula for an in terms of a0 and a1, and show the solution in the form y=a0y1(x)+a1y2(x) 2.what is the lower bound for the radii convergence...
  47. H

    Power Series for log z: Finding Singularity at z=0

    Homework Statement Does there exist a power series expansion of log z around z=0? If so, what is it? If not, demonstrate that it is impossible. Homework Equations [PLAIN]http://img839.imageshack.us/img839/4839/eq1.gif Here is the expansion of log z around z=1. The Attempt at a Solution...
  48. J

    Power series solution for differential equation

    Homework Statement Solve the fluxional equation (y with a dot on top)/(x with a dot on top) = 2/x + 3 - x^2 by first replacing x by (x + 1) and then using power series techniques.Homework Equations dy/dx = 2/x + 3 - x^2 The Attempt at a Solution First, I believe the fluxional (y with a dot...
  49. J

    Power series method and various techniques

    I know how to do problems like y' + y = 0 where you can replace y' and y with a series in sigma notation, manipulate and compare coefficients. But how do you solve a differential by power series that does not also include y or a higher order derivative? For example, y' = -(x^2) + 2/x + 3...
  50. A

    Power Series: Solve Arctan(x/sqrt(6)) Homework

    Homework Statement a) Determine the series of the given function. In the first box after the summation symbol, type in -1 or 1 indicating whether the series is alternating or not. b) Write out the sum of the first four nonzero terms of the series representing this function. c) Determine the...
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