What is Power series: Definition and 642 Discussions

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. C

    Non homogeneous ODE particular solution using power series

    Homework Statement Find the particular solution to the ODE y"+y=x using power series Homework Equations y=\sum(a_{n}x^{n})The Attempt at a Solution i tried plugging in y=\sum(a_{n}x^{n}) into the original equation and comparing coefficients of x to the first degree, but i am not sure how to...
  2. B

    Sketch on the complex plane the region where the following two power series both

    Homework Statement sketch on the complex plane the region where the following two power series both converge 1) sigma from n=0 to infinity [(z-1)^n]/[n^2] 2) sigma from n=0 to infinity [((n!)^2)((z+4i)^n)]/[2n]! The Attempt at a Solution R=lim as n tends to infinity...
  3. W

    Formation of the General Equation for a Power Series

    So I've gotten into the Method of Frobenius and all; Solved a few questions, however the most inconvenient part would be the formulation of the general equations for the final answer. Granted, the lecturer told us to not spend so much time on that segment due to its minimal weightage, but I...
  4. D

    Series solution up to a term, power series

    Homework Statement consider the initial value problem (1-x)y,,+xy,-2y=0 find the series solution up to the term with x6 Homework Equations (1-x)y,,+xy,-2y=0 The Attempt at a Solution assuming the answer has the form \Sigmaanxn that gives y,,=\Sigmananxn-1 and...
  5. Rapier

    Find Sum of Series Using Power Series

    Homework Statement Find the sum of sigma (n=1 to infinity) (-1)^(n-1) * [n/(4^(n-1))] using the power series 1/(1+x) = sigma (n=0 to infinity) (-1)^n * x^n. Homework Equations 1/(1+x) = 1 - x + x^2 - x^3 + x^4 + ... + (-1)^n * x^n The Attempt at a Solution The problem suggested...
  6. R

    Derivative and integral of a power series proof (not getting a step)

    Homework Statement Basically, there's no problem statement per se, I'm just trying to understand the proof that the following sequences have the same radius of convergence: (1) \displaystyle\sum_{k=0}^{\infty} c_{k}x^{k} (2) \displaystyle\sum_{k=0}^{\infty} kc_{k}x^{k-1} (3)...
  7. C

    Power Series Convergence for (5^n)(x-2)^n/8n^7: Explained with Ratio Test

    For the following power series: n=1 to infinity [(5^n)(x-2)^n]/8n^7 I used the ratio test, which I understand, but why does the book say it is convergent for 5|x-2|<7? I had 5|x-2|<1, but I don't understand why it would say 7?
  8. E

    Alternating Power Series - Limits

    1. Alternating power series question on convergence interval. I'm wrestling a bit with an alternating power series, the teacher has the convergence interval to be x = <-2,2] and I don't agree.Homework Equations Without further adue, here is the alternating power series in question...
  9. T

    Power Series Solutions and Radius of Convergence for y'' + xy = 0

    Homework Statement Find 2 independent solutions which are power series in x of y'' + xy =0 and find the radius of convergence of each solution. The Attempt at a Solution \sum_{n=2}^{\infty} n(n-1)a_n x^{n-2} + x\sum_{n=0}^{\infty}a_n x^n = 0 \sum_{n=-1}^{\infty} (n+3)(n+2) a_{n+3}...
  10. T

    How to Find Power Series Solutions of Chebyshev's Equation?

    Homework Statement Chebyshev's Equation is (1-x^2) y^{\prime\prime} - xy^{\prime} + c^2 y =0 where c is a real constant. (a) Find 2 linearly independent power series solutions of Chebyshev's Equation at x=0: an even one and an odd one. (b) Hence, using the ratio test, find the radius...
  11. T

    What is the power series solution to this differential equation?

    Homework Statement http://img59.imageshack.us/img59/2091/diffeq.png [PLAIN][PLAIN]http://img684.imageshack.us/img684/6748/diffeqp.png The Attempt at a Solution Making the substitutions y= \sum_{n=0}^{\infty} a_n x^n and y^{\prime} = \sum_{n=0}^{\infty}na_nx^{n-1}, \begin{align*}...
  12. M

    Representation of Functions as Power Series

    Find a power series representation for the function and determine the interval of convergence. f(x)=\frac{1+x}{1-x} This is one of the few problems in this section that I am getting stuck on. I know that I can relate it to the form of; \frac{1}{1-(x)} ...but after that I don't know what...
  13. A

    What is the Radius of Convergence for the Power Series Given?

    at the serie \sum_0^{\infty} a_n (x - c)^n , the radius of convergency is: . R= \lim_{n \to \infty } |\frac{a_n}{a_{n+1}}| My problem is : Find the radius of convergency when: \sum_0^{\infty} \frac{(-1)^n}{(2n+1)!} \cdot x^{2n+1} i don't understand mainly who is a_n . The...
  14. T

    Power Series Expansion Homework: Multiplication & n-k Addition Method

    Homework Statement I am doing this multiplication with power series and I am just stuck at this one and other questions that similar to this one. http://img5.imageshack.us/img5/9526/img1261r.jpg Homework Equations The Attempt at a Solution It seems that I suppose to add n-k...
  15. T

    Power Series: Interval Of Convergence

    Homework Statement I am not really good with Series so I having a hard time with these problems. http://img835.imageshack.us/img835/858/img1257d.jpg Homework Equations The Attempt at a Solution The part I am stuck is where I highlighted. The first question: The whole thing is squared so I...
  16. L

    Finding power series representation for ln(5-x).

    Homework Statement Find a power series representation for the function and determine the radius of convergence: f(x)=ln(5-x)Homework Equations Manipulate into the form 1/(1-x).The Attempt at a Solution I know how to do this with other functions, say, x/(9+x2)... It would convert to x/9 *...
  17. G

    Power series and the interval of convergence

    Homework Statement I need help finding the interval of convergence for f(x) = 3/(1-x^4). I think that the summation would be \Sigma 3 (x^4n) from n=0 to infinity, but I'm not sure how to get the interval of convergence. Homework Equations f(x) = 3/(1-x^4) The Attempt at a Solution...
  18. S

    Power series solution to a differential equation

    Homework Statement Using a power series solution, what is the solution to: (x^2-1)y" + 8xy' + 12y = 0 Homework Equations Normally these questions specify (about x0=0) but this one doesn't specify about which point. So if I use the power series equation, what am I supposed to plug in...
  19. Telemachus

    Finding power series for given values of a sum

    Homework Statement I have this exercise which I'm not sure how to solve. It says: Consider the series \displaystyle\sum_{0}^{\infty}x^n Does exists any value of x for which the series converges to five? ¿and to 1/3? Well, I've reasoned that if there exists that value, then it must be inside of...
  20. Telemachus

    Radius of Convergence for Power Series: What is the Limiting Ratio Test?

    Homework Statement Hi there. Well, I was trying to determine the radius and interval of convergence for this power series: \displaystyle\sum_{0}^{\infty} \displaystyle\frac{x^n}{n-2} So this is what I did till now: \displaystyle\lim_{n \to{+}\infty}{\left...
  21. U

    Asymptotic Power Series Behavior

    Homework Statement Suppose f(x) = \sum_1^\infty a_n x^n is a power series such that \lim a_{n+1}/a_n \to 1/n . Show that the magnitude of f(x) grows asymptotically as e^x . This is not a homework question. But, if I know why it is true (or if it is), then I can use it to answer a...
  22. I

    Converting Integral into Power Series: Is This Step Valid?

    Homework Statement Hi, we're supposed to put the following integral into a power series: \int \frac{arctan(t)}{t} dt with 0 < t < x. Homework Equations n/a The Attempt at a Solution I just want to know whether this step is ok. \int \frac{arctan(t)}{t} dt = \int \frac{1}{t} \int...
  23. P

    Finding the Function Represented by a Power Series

    Homework Statement Determine the an so that the equation \sum_{n=1}^{\infty}{na_{n}x^{n-1}} + 2\sum_{n=0}^{\infty}{a_{n}x^{n}} = 0 is satisfied. Try to identify the function represented by the series \sum_{n=0}^{\infty}{a_{n}x^{n}} = 0 Homework Equations The Attempt...
  24. C

    What is power series solution of y''-xy

    Homework Statement [PLAIN]http://img196.imageshack.us/img196/5241/recurrenceq.gif Homework Equations The Attempt at a SolutionThis is my attempted solution: 1) i got a recurrence relation (n+2)(n+1)a_(n+2)=a_(n-1) 2) i also used the matching coefficients method to get a2=0, a5=0, a7=0, but...
  25. L

    Power Series Approximation of an IVP

    1. Find the first four nonzero terms of the power series approximation of the solution. y"-4y = 4t-8e-2t y(0)=1, y'(0)=-1 2. y=\suma_n*t^n where the summation goes from 0 to infinity 3. I have done a homogeneous problem similar to this and had no problems finding the first four...
  26. M

    Radius/Interval Convergence for Power Series

    Homework Statement Find the radius and interval of convergence for the power series of n=0 of infinity of n^3(x-5)^n Homework Equations Ratio test: http://en.wikipedia.org/wiki/Ratio_test The Attempt at a Solution [(n+1)^3(x-5)^n+1 / n^3(x-5)^n] I am lost as to how to...
  27. A

    Power Series Expansion for 1/1+x

    Given that the sum of the geometric series is: 1+x+x^(2)+x^(3)+x^(4)...=1/1-x for -1<x<1 Find power series for 1/1+x Not to sure where to start, any help would be great
  28. T

    Power Series Problem: Am I Correct?

    Homework Statement Well, I'm not getting any problems from the question, but there is a part of the solution that I don't agree with. Here's the problem statement: http://img810.imageshack.us/img810/5387/powerseriesproblem.png Here's part of the solution...
  29. V

    Find the power series for representatino for the function

    Homework Statement Ive been able to do every single problem in my homework to the point where I have to test the edges of the interval of convergence. I have not been able to figure out a single one of the problems at the point of testing the edge of convergence, and I am to the point of...
  30. M

    Power Series - Interval of Convergence Problem

    Homework Statement For which positive integers k is the following series convergent? (To enter - or , type -INFINITY or INFINITY.) Summation of n=1 to infinity of (n!)^2 / (kn)! Homework Equations ratio test: limit n-->infinity of [((n+1)!)^2/(kn+1)!] / [(n!)^2 / (kn)!] (have the...
  31. V

    Question about power series

    Homework Statement find the power series for the function and determine the interval of convergence f(x) = {{x} \over {x^2 +1}} im trying latex for the first time so here it is if it doesn't show well: f(x) = x/(x^2 + 1)Homework Equations The Attempt at a...
  32. S

    Function represented by power series

    Homework Statement If a function f is represented by the power series ∑(k=0 to ∞) a_k(x-a)^k, with a radius of convergence )<R<∞, then f is continuous on the interval (a-R, a+R) Homework Equations The Attempt at a Solution I don't know if my proof is loose at some point or not...
  33. A

    Converting 1/(1+7x)^2 into a Power Series

    Homework Statement i need to make the function 1/(1+7x)^2 into a power series that goes from n=1 to infinity. I know that i have to get the answer through differentiating because the for the previous problem, i found that the function 7/(1+7x) resulted in ( -1 )^n* 7*7^n*x^n when n=0 to...
  34. J

    Power series solutions for ODEs. When are there how many of them?

    Hi, could someone please link me to the relevant theorems etc (or explain personally) that answer the issue that follows. Say you have an ODE (let's say 2nd order for now). Let's look for a power series solution (ie assume we're engineers). So, we write out a couple of sigmas etc and sub...
  35. K

    Help Convergence of Power Series, interval and radius of convergence question

    Homework Statement Determine the radius of convergence, the interval of convergence, and the sum of the series Summation from k=2 to ∞ of k(x-2)^k+1. Homework Equations ratio test? The Attempt at a Solution possibly take the derrivitive of the power series, then find the sum then integrate...
  36. M

    Show that the partial sums of a power series have no roots in a disk as n->infty

    Homework Statement Let f_n(z)=\sum_{k=0}^n\frac{1}{k!}z^n. Show that for sufficiently large n the polynomial f_n(z) has no roots in D_0(100), i.e. the disk of radius 100 centered at 0. Homework Equations This is a sequence of analytic functions which converges uniformly to e^z on C...
  37. S

    Power Series: Find Interval & Radius of Convergence

    Homework Statement \Sigma (from index k = 1 until infinity) Within the Sigma is the series : (k! * (x^k)) Homework Equations Ratio Test : lim as k approaches infinity |a(k+1) / ak| The Attempt at a Solution When I apply the ration test to the series and simplify I get lim k...
  38. M

    General formula for the sum of a finite power series

    I was wondering if there was a general way to find the sum of a finite power series: \sum_{n=1}^{N}{n^{m}} where m is a fixed integer. Now, there is some math folklore that a seven- (or ten-)year-old Gauss solved the m=1,\;N=100 case by realizing that by reversing the series and summing...
  39. B

    Using a Power Series to approximate a definite integral

    [PLAIN]http://img600.imageshack.us/img600/1210/11096142.png Hey I was wondering if you guys could help me out with this question... I think I have the right power series: = \frac{1}{1-x} + \frac{x}{1-x} = (1+x+x^{2}+x^{3}+...)(x+x^{2}+x^{3}+x^{4}+...) = 1+2x+2x^{2}+2x^{3}+2x^{4}+... = 1 +...
  40. jegues

    Power Series Recurrence Relation Problem

    Homework Statement See figure attached, we are asked to use power series to solve the differential equation. Homework Equations The Attempt at a Solution I'm confused as to how to deal with the -1 in the indices of one of my summations. I could add the term on the outside and...
  41. Saladsamurai

    Power Series: Question about constants

    Homework Statement So I have attached the problem in image: Pr and A are just numbers (constants) that are given. I solved the equation by power series solution. However, I am just confused because it is a second order DE but I only have one arbitrary constant of integration ao. I am not...
  42. C

    Finding the Radius & Interval of Convergence of a Power Series

    Homework Statement \sum from n=1 to inf (1+ 1/2 + ... 1/n)x^n Find the radius of convergence and the interval of convergence of the given power series. Homework Equations Dunno.. The Attempt at a Solution Stuck thinking about it. I'm not sure if I can combine what's in brackets with the...
  43. C

    Diff eq, power series solns, how do i determine how many terms to pull?

    Homework Statement find 2 power series solutions of the given diff eq about the ordinary point x = 0 y'' - xy = 0 Homework Equations y = (c_0)(y_1)[x] + (c_1)(y_2)[x] The Attempt at a Solution i can set it up to this (sorry idk out how to insert the subscripts with the summation symbols)...
  44. M

    Power series representation

    Homework Statement Find the power series representation for s(x) and s`(x) integral sin (pi t^2)\2 and which of them is valid ? Homework Equations The Attempt at a Solution I tried to solve this question , but i am not sure s`(x) = sin (pi t^2)\2...
  45. K

    Convergence of a Power Series: Lim Sup Explained

    Let sum(a_nx^n) be a power series with a_n not zero, and assume L=lim|a_(n+1)/a_n| exists. a) Show that if L is not zero, then the series converges for all x in (-1/L,1/L). b) Show that if L=0, then the series onverges for all x in R c) Show that a) and b) continue to hold if L is replaced by...
  46. M

    Prove that the power series for e^z does not converge uniformly on C.

    Homework Statement Prove that the power series for e^z does not converge uniformly on C. Homework Equations e^z=\sum_{k=0}^\infty z^k/k! The Attempt at a Solution The hint in the problem is to prove a proposition first: If f_n is a sequence of entire functions that converges...
  47. E

    Solution to an ODE using Power Series Method

    Homework Statement xy'-(x+2)y=-2x2-2x Homework Equations The Attempt at a Solution I'm clueless as to how to solve this as I'm only experienced in using the power series method with homogenous ODE's. Even if I make this homogenous, I don't know what to do with the x-variables that are not...
  48. H

    Power Series Odd Function Proof

    Homework Statement Suppose that f(x)= summation an x^n for n = 0 to infinity for all x. If f is an odd function, show that a0 = a2 = a4 = ... = 0.Homework Equations The Attempt at a Solution I said to consider sin(x), an odd function. When you do a series expansion only odd terms exist in the...
  49. J

    Power Series of Logs: Solving f(x)=x2ln(1-x)

    First post! I'm having a lot of trouble with power series, especially when there's more than one of the same variable in a function. Find the first few terms of the power series for the function f(x)=x2ln(1-x) I did a taylor series expansion of it, which gave me the right answer, but...
  50. K

    Power series representation of ln((1+2t)/(1-2t))

    Homework Statement Find a power series representation for the function f(t) = \ln((1+2t)/(1-2t)) Homework Equations f(t) = \ln((1+2t)/(1-2t)) The Attempt at a Solution \ln(1+2t)-\ln(1-2t) take derivative of f(t) expanded \frac{2}{1+2t}+\frac{2}{1-2t} 2 \int \frac...
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