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 & Interval of Convergence for Power Series

    Homework Statement Find the radius and interval of convergence for the following power series. \sum_{n = 2}^{\infty}\frac {(1 + 2cos\frac {\pi n}{4})^n}{lnn}x^n The Attempt at a Solution R = \frac {1}{\lim_{n\rightarrow\infty}\sqrt [n]{\frac {(1 + 2cos\frac {\pi n}{4})^n}{lnn}}}...
  2. C

    Power series of a function of 2 variables

    I have learned that if a function of one real variable can be defined as a power series, then this one is its Taylor series. Does the same occur with functions of 2 real variables? I mean, if a function f(x, y) can be defined as a power series, does this series is the Taylor series of f(x...
  3. B

    Power series representation of 10xarctan(5x).

    Homework Statement The function f(x)=10xarctan(5x) is represented as a power series http://img464.imageshack.us/img464/4131/formub5.jpg Find the first few(5) coefficients in the power series. Homework Equations I already know that the representation of arctanx is summation from...
  4. S

    Nonhomogeneous Power Series Solution

    For the fun of it, my DE book threw in a couple of problems involving nonhomogenous second order DE's in the section I'm currently going through. Although I have solved for the complementary solution, any suggestions on how to find the particular solution? For example, the one I'm looking at...
  5. M

    How to Find the Power Series and Radius of Convergence for Arctan'x

    how could i expand something such as arctan'x into a power series. also how would you be able to find the power series for it?so far i have managed to work out that: arctan'x = \frac{1}{1 + x^2} \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 +...+ (- 1)^n x^{2n} how do you work out the radius of...
  6. T

    What is the relationship between power series and exponential functions?

    1. I this from my homework solution. (1-t/s)^n = exp(-t/s) as n goes to infinity I don't understand. I checked the exponential power series. It should be : exp(x) = summation (x^n / n!) n=0 to infinity How come it could be a exponential function ? 2. another is that...
  7. S

    Power Series Solution of y''+(x^2)y=0: Is it Possible?

    y''+(x^2)y = 0 I tried to solve this problem using Power Series.But i can't make the solution in the form of series that have only two constants(a0,a1)that is, there are a0,a1, a2, a3. So i just wonder how can i make it has two constants.
  8. K

    Help with power series refinement

    Hi I stumbled across a power series pattern while working on a C algorithm and was wondering if this "discovery" has a name/reference anywhere. Basically, Here's what I found: for p = 1 the minimum number of terms, on the left side, satisfying the following is 1 a^p + b^p + c^p ... =...
  9. C

    Can someone help me find the power series representation for this function?

    I'm trying to do the question attached. I got the first three answers correct knowing that the nth derivative of a function evaluated at 0 divided by n! = c_n. However, I did the same for the others and the answer is incorrect. I know that I need the power series representation of that function...
  10. K

    What is the difference between power series and Taylor series?

    My exam is coming up, I have 2 questions on infinite series. Any help is appreciated!:smile: Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG For part a, I got: g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)] For part b, I got: x ∫ tan^-1...
  11. S

    Finding a power series representation

    Homework Statement Start with the power series representation 1/(1-x) = sum from n=0 to inf. of x^n for abs(x) < 1 to find a power series representation for f(x) and determine the radius of convergence. f(x)=ln(5+x^2) Homework Equations The Attempt at a Solution Okay, so I...
  12. K

    How Do You Find a Power Series Representation for f(x) = x / (4+x)?

    Homework Statement Find a power series representation for the function f(x) = x / (4+x) and determine the interval of convergence. I have no idea how to begin this problem. My only guess would be trying to divide something out in order to simplify to something that I'm able to...
  13. F

    Is This a Proper Power Series for Integration?

    Power series and integration >:( Hello, this question is about power series and integration of power series, the question and my working is on the image below. I had to write the question and my working plus the correct answer on a piece of paper and scan it, sorry for the hasle...
  14. S

    LaGrange Error and power series

    There's a homework problem that I've been struggling over: Find a formula for the truncation error if we use 1 + x^2 + x^4 +x^6 to approximate 1/(1-x^2) over the interval (-1, 1). Now, I assume that you need to use LaGrange error but I'm not sure how to proceed. Any help would be greatly...
  15. M

    Power Series for f(2x): Exploring the Expansion

    If f(x) has a power series: a_n(x-a)^n (centered at a) what does the power series for f(2x) look like?
  16. F

    Sum of Power Series Homework: Show Convergence and Determine Sum

    Homework Statement Show that the power series \sum_{k=1}^{k=\infty} \frac{x^{2k+1}}{k(2k+1)} converges uniformly when |x| \leq 1and determine the sum (at least when |x| < 1). The Attempt at a Solution Couldn't I somehow go about and show that, as |x| \leq 1, then f =...
  17. M

    Multiplying Power Series: Help & Solutions

    How do I multiply power series? Homework Statement Find the power series: e^x arctan(x) Homework Equations e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} arctan(x) = 0 + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} The Attempt at a Solution So do I multiply 1 by 0, x by...
  18. Repetit

    Solving xy'(x) - y(x) = x^2 Exp[x] using the power series method

    I've tried solving the equation xy'(x) - y(x) = x^2 Exp[x] using the power series method. I assume that y has the form: y = \sum_{n=0}^{\infty} a_n x^n Inserting this in the diff. eq. gives: \sum_{n=0}^{\infty} n a_n x^n - \sum_{n=0}^{\infty} a_n x^n = x^2 e^x Now, in the other...
  19. B

    Radius of convergence of power series

    Homework Statement Find the radius of convergence of the following series. \sum\limits_{k = 1}^\infty {2^k z^{k!} } Homework Equations The answer is given as R = 1 and the suggested method is to use the Cauchy-Hadamard criterion; R = \frac{1}{L},L = \lim \sup \left\{ {\left|...
  20. C

    Power Series & Singular Points: Why Change the Form?

    when finding a power series solution we have to put the differential equation ay''+by'+c=0 into the form y''+By+C=0 this leads to singular points when a=0 but why can't we leave the equation in its original form and use power series substitution to avoid singular points? or in...
  21. F

    Finding a Closed Form from a Power Series

    Homework Statement I have f(x) = the series x^k/[(k-1)k] summed from x=2 to infinity, and I need to find its closed form. Hint: What is the derivative of f(x) Homework Equations None. The Attempt at a Solution To start this problem, I took the derivative of the series, which...
  22. B

    Help with power series representation

    Homework Statement find a power series representation for the function and determine the radius of convergence. f(t)= ln(2-t) Homework Equations The Attempt at a Solution i first took the derivative of ln(2-t) which is 1/(t-2) then i tried to write the integral 1/(t-2)...
  23. C

    Power series and finding radius of convergence

    Homework Statement "Find the radius of convergence and interval of convergence of the series" \sum_{n=0}^\infty \frac{x^n}{n!} Homework Equations Ratio Test The Attempt at a Solution \lim{\substack{n\rightarrow \infty}} |x/n+1| (I can't seem to get the |x/n+1| to move up where it should be)...
  24. N

    Geometric power series of f(x)=6/(2-x) c=1

    Find the geometric power series for the given function: f(x)=6/(2-x) c=1 I am stumped on this one. I've tried for an hour on this one with no luck. Could someone help?
  25. S

    Finding the Sum of a Power Series

    I'm trying to find the sum of this: \[ \sum\limits_{n = 0}^\infty {( - 1)^n nx^n } \] This is what I have so far: \[ \begin{array}{l} \frac{1}{{1 - x}} = \sum\limits_{n = 0}^\infty {x^n } \\ \frac{1}{{(1 - x)^2 }} = \sum\limits_{n = 0}^\infty {nx^{n - 1} } =...
  26. S

    Radius / Interval of Convergence (Power Series)

    I need help finding the radius & interval of convergence of the following power series: The sum from n=0 to infinity of... (2-(n)^(1/2)) * (x-1)^(3n) I think the ratio test is supposed to work, but I narrow the limit of the test down to 1/|x-1|^3 < 1, and this doesn't make sense to me. Is...
  27. F

    Maclaurin Power Series for 1/(4x^2+1) and Integration of e^-x^2

    I was hoping someone could check my work: Find the maclaurin power series for the function: a. f(x)=1/(4x^2+1) b. f(x)= \int e^-x^2 dx For a I got (-1)^n*2nx^n. For b I don't know where to start.
  28. F

    Geometric Power Series Representation of ln(1+2x) at c=0

    I was wondering if someone could check my work: Find the geometric power series representation of f(x)=ln(1+2x), c=0 I get \\sum_{n=0}^ \\infty2(-2x)^n+1 on -1/2<x<1/2
  29. T

    Solving a Power Series Question: X = yt/2(sinh yt + sin yt/cosh yt - cos yt)

    another power series question... I tried it for awhile but it just got out of hand and the amount of numbers got unbearable. Q. The increase in resistance of strip conductors due to eddy currents at power frequencies is given by : X = yt divided by 2 (sinh yt + sin yt divided by cosh yt -...
  30. M

    Expansion of y=sinh-1(x) in terms of x using inversion of power series method

    I need to express the sinh-1(x) as a power series in terms of powers of x. I have written the expression as x=sinhy and expanded the sinhy using the exponential series to give x = y+(1/3)y^3+(1/5)y^5+... I guess I need to expand the y=sinh-1(x) and compare or equate the coefficients. If this...
  31. K

    Calculating Power Series Coefficients with Differentiability

    Let be the powe series: f(x)=\sum_{n=0}^{\infty}a(n)x^{n} then if f(x) is infinitely many times differentiable then for every n we have: n!a(n)=D^{n}f(0) (1) of course we don't know if the series above is of the Taylor type, but (1) works nice to get a(n) at least for finite n.
  32. H

    Power series to solve 2nd order ordinary differential equations

    I need some help with power series. I can't remember how to find a power series center around a point. example question: y"-xy'-y=0, x=1 I don't how to start this.
  33. S

    Complex Functions (Power Series)

    Hi, Power Series' were not covered in my cal II class, so I don't know how to solve these. Is there a certain way to solve these? Find the Radiusof convergence and open disk of convergence of the power series: \frac{n^2}{2n+1}(z+6+2i)^n I don't know how to latex the summation but it is...
  34. E

    Solve Recurrence Relation for DE x(x-2)y''+(1-x)y'+xy=0@x=2

    Im trying to determine the recurrence relation of the following differential equation : x(x-2)y'' + (1-x)y' + xy =0 about the regular singular point x =2. I've tried rewriting the DE as (x-2+2)(x-2)y'' -(x-2+1)y' +(x-2+2)y =0, but it doesn't seem to work. any ideas?
  35. C

    What can be said about the convergence or divergence of the following series?

    Suppose that \sum_{n=0}^{\infty} c_{n}x^{n} converges when x=-4 and diverges when x=6 . What can be said about the convergence or divergence of the following series? (a) \sum_{n=0}^{\infty} c_{n} (b) \sum_{n=0}^{\infty} c_{n}8^{n} (c) \sum_{n=0}^{\infty} c_{n}(-3)^{n} (d)...
  36. P

    Expand Denominator of $\frac{x^3}{e^x-1}$ as Power Series

    I have a function, \frac{x^3}{e^x-1} The question than says expand the denominator as a power series in e^{-x}.I don't understand this question. How do I start doing that? It is not suggesting to approximate \frac{1}{e^x-1} as e^{-x} is it?
  37. I

    Simple question regarding formulae for power series

    Sorry if this is in the wrong forum, I think it's right but I'm not totally sure. I'm (slowly) working my way through Roger Penrose's brilliant book Road to Reality. Recently I finished chapter four (Magical Complex Numbers) and one thing had me confused. How does he go about getting those...
  38. siddharth

    Can Power Series Solve the Differential Equation xy' - 3y = k?

    Question: Find a power series solution in powers of x for the following differential equation xy' - 3y = k My attempt: Assume y = \sum_{m=0}^{\infty} a_m x^m So, xy' = \sum_{m=0}^{\infty}m a_m x^m xy'-3y-k=0 implies \sum_{m=0}^{\infty}m a_m x^m - 3\sum_{m=0}^{\infty} a_m x^m - k...
  39. D

    Non-homogeneous 2nd order diff eq involves power series

    I just need a hint or something to see where I start. I'm at a loss for a beginning. Consider the non-homogenous equation y'' + xy' + y = x^2 +2x +1 Find the power series solution about x=0 of the equation and express your answer in the form: y=a_0 y_1 + a_1 y_2 + y_p where a_0 and...
  40. D

    Non-Homogenous Power series 2nd order equation

    I just need a hint or something to see where I start. I'm at a loss for a beginning. Consider the non-homogenous equation y'' + xy' + y = x^2 +2x +1 Find the power series solution about x=0 of the equation and express your answer in the form: y=a_0 y_1 + a_1 y_2 + y_p where a_0 and...
  41. H

    HELP: A complex valued power series

    Hi All, I have this here power series which is complex valued \frac{2n+1}{2^n} i^n = \frac{4}{25} + \frac{22}{25}i My task is to prove that this power series has the above mentioned sum. To do this I separate the sum into two sums S_0 + S_1 = (i/2)^n + 2* (i/2)*n*(i/2)^{(n-1)}...
  42. K

    How Can Power Series Solutions Simplify Differential Equations?

    Given: y'+2xy=0 Find: Write sereis as an elementary function My solution so far: y=[Sum n=0, to infinity]C(sub-n)*x^n y'=[Sum n=1, to infinity]n*C(sub-n)*x^(n-1) y' can be transformed into: =[Sum n=0, to infinity](n+1)*C(sub-n+1)*x^n ([Sum n=0, to infinity](n+1)*C(sub-n+1)*x^n)...
  43. E

    Power Series Solution to Hydrogen Wave Function Differential Equation

    I solved the differential equation for theta portion of the hydrogen wave function using a power series solution. I got a sub n+2 = a sub n ((n(n+1)-C)/(n+2)(n+1)). I then truncated the power series at n = l to get C= l(l+1). I know need to use the recursion formula I found to find the l =...
  44. B

    Evaluate Limit Using Taylor Approximation of Power Series of e^h

    Use a four-term Taylor approximation for e^h , for h near 0 , to evaluate the following limit. lim (e^h-1-h-h^2/2)/h^3 h->0 i know that e^h = 1+h+h^2/2+h^3/3+h^4/4... therefore, I say that e^h-1-h-h^2/2 = h^3/3+h^4/4... (h^3/3+h^4/4...)/h^3 is approximately = 1/3 but its wrong...
  45. D

    Shifting index of summation of power series

    I can't seem to get these power series to match up so that I can solve the equation...heres my work:
  46. Z

    Power series differential equation

    This is an initial value problem. y''=y'+y, y(0)=0 and y(1)=1 I am solving it by deriving the Fibonnacci power series. The sum of F_n from n=0 to infinity is 0,1,1,2,3,5,8,13 of Fibonnacci numbers defined by F_o = 0, F_1 =1. F_n = F_n-2 + F_n-1 for n>1. Here is what I have attempted so far...
  47. M

    Stuck on power series, need a refreshment Diff EQ converge?

    Hello everyone! I remember doing these in calc II, but forgot 99% of it. Here is the question: Find the interval of convergence for the given power series. http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/5e/119a4cd07ce5d3f6f673477ed169641.png The series is convergent from x = ...
  48. T

    Finding Power Series Representation for arctan(x): Help Needed!

    may i know how to solve this ques:find the power series representation for arctan (x) i know that arctan (x) = integ 1/(1 + x^2) but then from here i don't know how to continue. pls help...
  49. B

    Solving a DE: Finding Power Series Solutions about z = 0

    Can someone please explain some steps of a worked example. Q. Find the power series solutions about z = 0 of 4zy'' + 2y' + y = 0. (note: y = y(z)) Writing the equation in standard form: y'' + \frac{1}{{2z}}y' + \frac{1}{{4z}}y = 0 Let y = z^\sigma \sum\limits_{n = 0}^\infty...
  50. T

    Power series to find the inverse of any function in Z_2[x]?

    Is it possible to use power series to find the inverse of any function in Z_2[x]?
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