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. K

    Power series representation question

    Homework Statement Determine the value of f(-1) when Homework Equations f(x) = (x/2^2) + ((2x^3)/2^4)+((3x^5)/2^6)+... . (Hint: differentiate the power series representation of ((x^2)-2^2)^(-1).) The Attempt at a Solution I was not very sure were to begin on this one. So I...
  2. R

    Complex exponential proof using power series

    I need to prove that ez1 x ez2 = e(z1 + z2) using the power series ez = (SUM FROM n=0 to infinity) zn/n! (For some reason the Sigma operator isn't working) In the proof I have been given, it reads (SUM from 0 to infinity) z1n/n! x (SUM from 0 to infinity)z2m/m! = (SUM n,m)...
  3. H

    Power Series Interval of Convergence

    Homework Statement I need a power series with a radius = pi. (So when you do the ratio test on this power series you get pi) Homework Equations The Attempt at a Solution I tried x^n*sin(n) and thought of stuff like that but couldn't come up with a working power series
  4. M

    Using power series remainder term

    Homework Statement (For power series about x=1) Using the error formula, show that \left|ln(1.5)-p_{3}(1.5)\right|\leq\frac{(0.5)^{4}}{4} Homework Equations p_{3}(x) = x-1 - \frac{(x-1)^{2}}{2} + \frac{(x-1)^{3}}{3} \\\epsilon_{n}(x)=\frac{f^{n+1}(\xi)}{(n+1)!}(x-x_{o})^{n+1}\\where \xi lies...
  5. S

    Differentiation of power series

    Homework Statement Show that 4 = \sum from n = 1 to \infty (-2)^{n+1} (n+2)/n! by considering d/dx(x^{2}e^{-x}). Homework Equations Power series for e^{x} = \sum x^{n}/n! from 0 to \infty. The Attempt at a Solution So I started with the power series for e^{-x} = \sum -x^{n}/n...
  6. E

    Do we always bump the bound up to n=1 when differentiating a power series?

    My question is just a concept that I don't understand. When differentiating a power series that starts at n=0 we bump that bound up to n=1. My question is do we always do that? or Do we only do that when the first term of the power series is a constant and thus when it is...
  7. K

    Power series solutions to differential equations

    Homework Statement I'm revising at the moment and a bit stumped on question 4 http://www.maths.ox.ac.uk/system/files/attachments/AC104.pdf Homework Equations The Attempt at a Solution I think for the first part of the question, the regular singular points are 0 and -2...
  8. G

    Uses of power series as opposed to taylor series

    So we can use the Taylor's theorem to come up with a Taylor series represent certain functions. This series is a power series. So far (I'm in my second year of calc, senior in high school), I've never seen a power series that wasn't a Taylor series. So are all power series taylor series? Whether...
  9. J

    Legendre differential equation- power series

    Homework Statement http://mathworld.wolfram.com/LegendreDifferentialEquation.html I have a question about how the website above moves from one equation to another etc. 1./ Equations (4), (5) and (6) When differentiating (4) to (5) shouldn't the the limit be from n=1, which means (5)...
  10. D

    Power Series Representation of a Function Help

    [b]1. Homework Statement : Find the power series for the function f(x)=5/(7-x), centered at c=-2. [b]2. Homework Equations : a/(1-r) [b]3. The Attempt at a Solution : I know that I need to divide by seven to get (5/7)/(1-(x/7)) and then rewrite in the form the sum of (a)(r)^n. I tried adding 2...
  11. H

    Coefficient of the product of two power series

    If a_0 + (a_1)x + (a_2)x^2 + ... and b_0 + (b_1)x + (b_2)x^2 + ... are two power series and the coefficient of x^r from their product is a power series: (a_0)(b_r) + (a_1)(b_(r-1)) + ... What principle or theorem or definition(s) are we applying when finding that this is indeed the...
  12. C

    Expanding into Power Series (Complex)

    Homework Statement Expand e^{1/z}/\sin z in powers of z+1+i.Homework Equations Not sure, see below.The Attempt at a Solution I already know that \begin{align} \sin z & = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} \end{align} And the other expansion for the exponential (but we just...
  13. C

    Power Series Expansion about Point

    So this is a REALLY elementary question but I can't seem to find the answer on the net, or maybe I did but just keep skipping over it some how. (by the way, this is with respect to complex numbers z \in C which is used in Complex Analysis, thus why I chose this forum). I know what it means...
  14. C

    Represent (1+x)/(1-x) as a power series.

    Homework Statement Represent (1+x)/(1-x) as a power series. Homework Equations The Attempt at a Solution I started with 1/ (1-x) = sum (x)^n n= 0 - infinity (1 + x) sum x^n and this is where I am stuck.Homework Statement Homework Equations The Attempt at a Solution
  15. D

    ( ) Power series solution for ODE

    (URGENT) Power series solution for ODE Homework Statement Supose there is an infinite series solution\sum b_{n}x^{n} for u''+4(x-(1/4))^2*u+C(x) = 0 where C(x) is a function (I get it in another problem, I'll put it in the relevant equations area), determinate the coefitients b_{0} b_{1}...
  16. S

    Express the indefinite integral as a power series.

    Hello, I'm kind of stuck in this problem. I have to express the integral as a power series. the integral of (e^x -1)/x I thought about evaluating it as f(x)=(e^x -1)/x and treating it as a Taylor series is that correct? Could I have any other hints? I would really appreciate it...
  17. A

    Power Series Representation of a Function when a is a polynomial

    Power Series Representation of a Function when "r" is a polynomial Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x)=\stackrel{(1+x)}{(1-x)^{2}} Homework Equations a series converges when |x|<1...
  18. I

    Non-linear dynamics approach to a manifold of a saddle point using power series

    Homework Statement Im taking a dynamics course and I am using The strogatz book Non-linear Dynamics and Chaos I need to solve a problem that is similar to problem 6.1.14 Basically it consist in the following You have a saddle node at (Ts,Zs) which is (1,1). Consider curves passing through...
  19. A

    Power series: How would you write this off as?

    So. There's this question about power series that will eventually take the form of p= |x| lim n->inf | nn / (n+1)(n+1) | But of course, in a futile attempt at a solution I tried doing the derivative for both functions. Didn't get anywhere of course. Knowing that eventually the answer...
  20. A

    Power Series Calc 2: Determine 1/(1+9x)^2 From n=1 to ∞

    Homework Statement Determine the power series for g(x)=1/(1+9x)^2 The sigma in the answer has to be from n=1 to infinity We also have to specify whether it is alternating by putting either (1)^n or (-1)^n This is an online problem and I have no idea why what I am putting is not right...
  21. M

    Rewriting Power Series - Simple Algebra Question

    Homework Statement My question involves a small algebra issue within a power series problem. I have (-1)^n-1 and i just need to know how i can re-write this. I know that if it were (1)^n+1 i could re-write as (1)^n * (n) So can i write (-1)^n-1 as (-1)^n * (-1) ? Homework Equations...
  22. J

    Frobenius power series repeated roots

    Could someone please explain the y2 solution for repeated roots in Frobenius method where y2=y1lnx+xs \Sigma CnxnI am struggling to figure out how to solve this
  23. Saladsamurai

    Power Series Change of Indices: I broke math again

    Homework Statement I have an infinite series that looks like this: \sum_0^\infty n(n-1)d_nx^{n-1} + \sum_0^\infty n(n-1)d_nx^n + \sum_0^\infty d_nx^nI wish to combine all three sums so that they must all have same powers of x and start at same index. The second and third summations are fine...
  24. Saladsamurai

    Diff EQs: Power Series vs Frobenius Solutions: Difference?

    This is a pretty general conceptual question. I was just doing some reviewing for a test, and it occurred to me that if I were not told specifically to use Frobenius method on an equation, I might try to Power series solve it and vice versa. Can we talk about the difference a bit? We apply...
  25. J

    Power series when to use Frobenius method

    Hi, I'm new to the forum and need some help regarding my calc class. Any help you could provide would be greatly appreciated. In doing a power series series solution when should I use the frobenius method and when should I use the simple power series method. The simple method seems a little...
  26. S

    Power series expansion for Log z

    Homework Statement Find the power series expansion of Log z about the point z = i-2. Show that the radius of convergence of the series is R = \sqrt{5}. Homework Equations None The Attempt at a Solution I know that Log z = (z-1) - (1/2)(z-1)^2 + (1/3)(z-1)^3 -... So wouldn't this...
  27. Saladsamurai

    Power Series Solution for x^2y'' - y = 0 Expanding about xo = 2: Next Steps?

    Homework Statement Solve x^2y'' - y = 0 using Power Series Solution expanding about xo = 2. The Attempt at a Solution First I expand the coefficient of y" (i.e. x2) about xo: TS[x^2]|_{x_o=2} = 4+ 4(x - 2) + (x - 2)^2 Assuming the solution takes the form: y(x) = \sum_0^{\infty}a_n(x -...
  28. Saladsamurai

    Power Series Following an example problem

    Homework Statement I am following along in an example problem and I am getting hung up on a step. We are seeking a power series solution of the DE: (x - 1)y'' + y' +2(x - 1)y = 0 \qquad(1) With the initial values y(4) = 5 \text{ and }y'(4) = 0. We seek the solution in the form y(x) =...
  29. jegues

    Taylor Series using Geometric Series and Power Series

    Homework Statement See figure attached. Homework Equations The Attempt at a Solution Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the, \frac{1}{x^{2}} \int x^{-2} = \frac{-1}{x} + C How do I...
  30. jegues

    Exploiting Geometric Series with Power Series for Taylors Series

    I'm confused between some formulae so I'm going to give some examples and you can let me know if what I'm writing is correct. Find the Taylor series for... EXAMPLE 1: f(x) = \frac{1}{1- (x)} around x = 2 Then, \frac{1}{1-(x)} = \frac{1}{3-(x+2)} = \frac{1}{3} \left( \frac{1}{1...
  31. S

    For the following power series: ∑ (4^n x^n)/([log(n+1)]^(n)

    For the following power series, find ∑ (4^n x^n)/([log(n+1)]^(n) (a) the radius of convergence (b) the interval of convergence, discussing the endpoint convergence when the radius of convergence is finite. -------------------------------------------------------------------------------...
  32. B

    Is Power Series Convergence Related to Other Series Convergence?

    Homework Statement If \sum_{n=0}^{\infty} c_{n}4^n is convergent, does it follow that the following series are convergent? a) \sum_{n=0}^{\infty} c_{n}(-2)^n b) \sum_{n=0}^{\infty} c_{n}(-4)^n Homework Equations The Power Series: \sum_{n=0}^{\infty} c_{n}(x - a)^n The...
  33. S

    Uniqueness theorem for power series

    Hi, for awhile I was agonizing over part b) of this http://books.google.com/books?id=WZX4GEpxPRgC&lpg=PP1&dq=lang%20complex%20analysis&pg=PA62#v=onepage&q&f=false" of Theorem 3.2 in Lang's Complex Analysis. But I think part of the reason was that I kept concentrating on the second sentence...
  34. L

    1 last infinite series, power series

    Homework Statement suppose a large number of particles are bouncing back and forth between x=0 and x=1, except that at each endpoint some escape. Let r be the fraction of particles reflected, so then you can assume (1-r) is the number of particles that escape at each wall. Suppose particles...
  35. L

    Infinite series, power series problem

    Homework Statement In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if n=2, the water can be made a pure as you like, but if n=3, at...
  36. P

    How Is the Tan Power Series Derived Using Sin, Cos, and Bernoulli Numbers?

    How can the tan power series be derived from the sin and cos power series? Where do the Bernoulli numbers come in?
  37. P

    On the radius of convergence of a power series

    Hi, I'm new here. I am curious that why a power series must have a radius of convergence? I mean, even in a complex plane, there is always a so-called convergent radius for a power series. Is it possible that a power series is convergent for a certain range in one direction, and for an apparent...
  38. W

    Convergence of Power Series without Recursion Relation

    Homework Statement Suppose I have the power series: f(x) = A0 + A1 x +A2 x^2 ...An x^n Where A0..An are numbers, there is no recursion relation. Find the interval of convergence Homework Equations The Attempt at a Solution Can I use ratio test? How would I do this since there is no recursion...
  39. I

    Function represented as a power series

    Homework Statement http://img.photobucket.com/albums/v257/gamer567/powerseries.png Homework Equations The Attempt at a Solution http://img.photobucket.com/albums/v257/gamer567/cramster.png I am getting lost from the transition in the 1/x-2 to 1/(x-2)^2. Could someone...
  40. M

    Power Series Representation for Arctan(x)

    Homework Statement F(x)=∫(0 to x) tan^(-1)t dt. f(x)= infinite series ∑n=1 (-1)^(en)(an)x^(pn)? en=? an=? pn=? I know en = n-1 Homework Equations The Attempt at a Solution Start with the geometric series 1/(1 - t) = ∑(n=0 to ∞) t^n. Let t = -x^2: 1/(1 + x^2) = ∑(n=0 to ∞)...
  41. Z

    Asymptotic formula for a power series

    where can i find a proof of the following identity ? \sum_{n=0}^{\infty} (-x)^{n} \frac{c(n)}{n!} \sim c(x) +(1/2)c''(x)x+(1/6)c'''(x)x + (1/8)x^{2}c'''' (x) +++++
  42. estro

    Defining y and Exploring Properties of Power Series

    \mbox {Suppose I have: } \sum_{n=1}^\infty (\frac {x} {3})^{2n} \mbox{Can I define } y= \frac {x} {3} a_k(y) = \left\{ \begin{array}{c l} (y)^k, & \mbox{if } k= 2n\\ \\ (0)^k, & \mbox{otherwise} \end{array} \right. \mbox {And then use all the cool properties of power series on }...
  43. N

    Conditional Convergence in a Power Series

    I was wondering if there's an example of a power series \sum_n^\infty c_n (z-a)^n with radius of convergence R so that all z for which |z-a| = R there is purely conditional convergence? (no divergence but also no absolute convergence) Or perhaps a reason why that's impossible?
  44. S

    Power Series Expansion and Residue Calculation for log(1-z)

    Homework Statement Find a power series expansion for log(1-z) about z = 0. Find the residue at 0 of 1/-log(1-z) by manipulation of series, residue theorem and L'Hopitals rule. Homework Equations The Attempt at a Solution Is this power series the same as the case for real numbers.
  45. R

    Complex number and power series

    Homework Statement Let ω be the complex number e^(2πi/3)， Find the power series for e^z + e^(ωz) + e^((ω^2) z). Homework Equations The Attempt at a Solution I can show that 1+w+w^2=0, don't know if it would help. Could anyone please give me some hints? Any input is appreciated!
  46. N

    Where can I find videos on power series and other calculus topics?

    Hey All, I'm learning calculus through videos. The videos that I'm watching are really good, but they are deficient in power series, taylor and mclaurin series, binomial series, and taylor polynomial applications. Anyone know where I can see video instruction for these? Steve
  47. R

    How Do You Differentiate a Power Series Like \(\sum \frac{x^n}{n \cdot 3^n}\)?

    Consider the power series (n=1 to infinity) \Sigma (x^n)/(n*3^n). (a) Find the radius of convergence for this series. (b) For which values of x does the series converge? (include the discussion of the end points). (c) If f(x) denotes the sum of the series, find f'(x) as explicitly as...
  48. F

    Power Series for ODE: Find Coefficient of x38 Term

    Homework Statement Find the first 6 terms of the power series expansion centered at 0 for the general solution for y -xy'=0. Then find the coefficient of the x38 term. Homework Equations General solution is of the form: y=a0+a1x+a2x2+a3x3+a4x4+a5x5+... If you factor out the ao and...
  49. N

    Converting Power Series to Integrals: How to Handle Constants of Integration?

    \int \frac{x-arctanx}{x^3}dx \frac{d}{dx}( x-arctanx ) = 1-\frac{1}{1+x^2}=\frac{x^2}{x^2+1} = x^2 \sum_{n=0}^{\infty}(-1)^nx^{2n} = \sum_{n=0}^{\infty}(-1)^nx^{2n+2} \int \sum_{n=0}^{\infty}(-1)^nx^{2n+2} dx = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+3}}{2n+3}+C C=0? \int...
  50. P

    How Do You Calculate the Coefficients and Convergence of a Complex Power Series?

    Homework Statement Suppose that f(z) = ∑a_j.z^j for all complex z, the sum goes from j=0 to infinity. (a) Find the power series expansion for f' (b) Where does it converge? (c) Find the power series expansion for f^2 (d) Where does it converge? (e) Suppose that f'(x)^2 + f(x)^2 = 1...
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