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. yecko

    Solving a Linear ODE using a power series

    Homework Statement Homework Equations Power series ODE The Attempt at a Solution [/B] Sorry for not typing all those things out from my phone.. How can I get C1? And how can I put the solution in the required format? (I don't know how to put it in summation sign... and i cannot even solve...
  2. yecko

    2nd order differential equation with power series

    Homework Statement Homework Equations Power series The Attempt at a Solution As I have to write in form of "x^2n" & "x^2n+1", I am totally have no idea with how can I go on to do the question. Those I have learned in lecture and online are mostly with only one part of summation... or two...
  3. C

    Coefficient Matching for different series

    Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...
  4. N

    Power Series Equation for Amplifier and Harmonics

    Hi, I keep reading in multiple sources that amplifier output can be given by Vout = a0 + a1v(t) + a2v2(t) + a3v3(t) + ... + anvn(t) I've checked in three of my textbooks and there is not a clear definition (its often just stated) why this equation is used and why it works. I am not looking...
  5. T

    MHB Power Series Convergence Assistance

    The power series $$\sum_{n = 2}^\infty \frac{(n-1)(-1)^n}{n!}$$ converges to what number? So far, I've tried using the Ratio Test and the limit as n approaches infinity equals $0$. Also since $L<1$, the power series converges by the Ratio Test.
  6. J

    Finding the singular points for this differential equation

    Homework Statement If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations included in attempt The Attempt at a Solution I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though. Whenever I expand for m=0 for the a0 I end...
  7. T

    Can Entropy Be Expressed as a Power Series in Terms of Internal Energy?

    Is it ok to assume that the entropy ##S## of an arbritary system can be written as a power series as a function of the system's internal energy ##U##? Like $$S(U) = \sum_{i=1}^{\infty}a_iU^i = a_1 U + a_2 U^2 + \ ...$$ with ##a_i \in \mathbb{R}##. What results could be obtained from such...
  8. P

    Expressing series in terms of a Power Series

    Hello and thank you for trying to help. In spite of the fact that this seems a very simple problem, I do not find myself able to get a solution. Here it goes: Let $$f(x)=\displaystyle \sum_{k=3}^\infty a_k \frac{x^k}{k(k-1)(k-2)}$$ and $$g(x)=\displaystyle \sum_{k=0}^\infty a_k x^k$$. Express...
  9. jaketodd

    I Cardinality of the Power Series of an Infinite Set

    According to this page: https://en.wikipedia.org/wiki/Cantor's_theorem It says: "Cantor's theorem is a fundamental result that states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself." Furthermore, it says: "Cantor's...
  10. karush

    MHB Find Power Series Representation for $g$: Interval of Convergence

    $\textrm{a. find the power series representation for $g$ centered at 0 by differentiation}\\$ $\textrm{ or Integrating the power series for $f$ perhaps more than once}$ \begin{align*}\displaystyle f(x)&=\frac{1}{1-3x} \\ &=\sum_{k=1}^{\infty} \end{align*} $\textsf{b. Give interval of convergence...
  11. M

    MHB Power series and uniform convergence.

    Hi. I have this power serie (2^n/n)*z^n that runs from n=1 to infinity, and I have to show whether it's uniform konvergence on [-1/3, 1/3] or not. I hope someone can help me with this.
  12. PhysicsCollegeGirl

    Master Power Series Convergence with Expert Help - Examples Included

    Homework Statement [/B] There are three problems that I am struggling with. 1. ∑[k2(x-2)k]/[3k] 2. ∑[(x-4)n]/[(n)(-9)n] 3. ∑[2k(x-3)k]/[k(k+1)] The Attempt at a Solution On the first two I am having problems finding the end-points of the interval of convergence. I use the ratio test. 1...
  13. Eclair_de_XII

    Finding a power series solution to a differential equation?

    Homework Statement "Find the recurrence relation in the power series solution for ##y''-xy'-y=0## centered about ##x_0=1##." Homework Equations ##y=\sum_{n=0}^\infty a_nx^n## Answer as given in book: ##(n+2)a_{n+2}-a_{n+1}-a_n=0## The Attempt at a Solution ##y=\sum_{n=0}^\infty a_n(x-1)^n##...
  14. Mr Davis 97

    Proving the Power Series Expansion of 1/(1+x^2)

    Homework Statement Show that ##\displaystyle \frac{1}{1+x^2} = \frac{1}{x^2} - \frac{1}{x^4} + \frac{1}{x^6} - \frac{1}{x^8} + \cdots## Homework EquationsThe Attempt at a Solution I know that the power series expansion of ##\displaystyle \frac{1}{1+x^2}## about ##x=0## is ##1-x^2 + x^4 - x^6 +...
  15. uchuu-man chi

    Need help evaluating an improper integral as a power series.

    Homework Statement Evaluate the indefinite integral as a power series. What is the radius of convergence (R)? ##\int x^2ln(1+x) \, dx## Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (-1)^n \frac {x^{n+3}} {n(n+3)}; R = 1## Homework Equations Geometric series ##\frac {1} {1-x} =...
  16. N

    I Power series - Different problem

    In the power series below, I've used the ratio test and at the end I got |x-2| times infinity which is >1 so it diverges.. and in this case there is no interval of convergence because it's times inifnity.. How did he conclude that it converges at x=2??
  17. N

    I Absolute Power Series: Questions & Solutions

    I've 2 questions 1) Why do we take absolute of the power series? 2) I don't get why the interval of convergence is from -inifinity to +infinity. You can find the problem below.
  18. JamesonS

    Solve Diff. Eq. using power series

    Homework Statement \begin{equation} (1-x)y^{"}+y = 0 \end{equation} I am here but do not understand how to combine the two summations: Mod note: Fixed LaTeX in following equation. $$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$
  19. ExplosivePete

    Power Series expansion of an eigenvalue

    1. ... Expand the Eigenvalue as a power series in epsilon, up to second order: λ=[3+√(1+4 ε^2)]V0 / 2 Homework Equations I am familiar with power series, but I don't know how to expand this as one.[/B]The Attempt at a Solution :[/B] I have played around with the idea of using known power...
  20. M

    Find the power series in x-x0?

    Homework Statement Find the power series in x-x0 for the general solution of y"-y=0; x0=3. Homework Equations None. The Attempt at a Solution Let me post my whole work:
  21. B

    Radius of Convergence for Ratio Test in Calculus Questions

    Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...
  22. karush

    MHB 206.r2.11find the power series representation

    $\tiny{206.r2.11}$ $\textsf{find the power series represntation for $\displaystyle f(x)=\frac{x^7}{3+5x^2}$ (state the interval of convergence), then find the derivative of the series}$ \begin{align} f(x)&=\frac{x^7}{3}\implies\frac{1}{1-\left(-\frac{5}{3}x^2\right)}&(1)\\...
  23. karush

    MHB 206.11.3.12 write the power series

    $\textsf{a. Find the first four nozero terms of the Maciaurin series for the given function} \\$ \begin{align} f^0(x)&=\ln{ (6 x + 1)} &\therefore f^0(a)&=0\\ f^1(x)&=\frac{6}{(6 x + 1)} &\therefore f^1(a)&=6\\ f^2(x)&= \frac{-36}{(6 x + 1)^2} &\therefore f^2(a)&=-36\\ f^3(x)&= \frac{432}{(6 x...
  24. F

    Integral x/(1-x) via power series?

    So, ∫x/(1-x)... can I solve this as a power series ∫(x*Σ x^n) = ∫(Σ x^(n+1))= (1/(n+2)*Σ x^(n+2))? Is this correct? I know there is other ways to do it... But should this be correct on a test? This solution is more fun..
  25. karush

    MHB S4.12.9.13 find a power series representation

    $\tiny{s4.12.9.13}$ $\textsf{find a power series reprsentation and determine the radius of covergence.}$ $$\displaystyle f_{13}(x) =\frac{1}{(1+x)^2}=\frac{1}{1+2x+x^2}$$ $\textsf{using equation 1 }$ $$\frac{1}{1-x} =1+x+x^2+x^3+ \cdots =\sum_{n=0}^{\infty}x^n \, \, \left| x \right|<1$$...
  26. Battlemage!

    Using substitution to turn a series into a power series.

    Homework Statement The problem asks to use a substitution y(x) to turn a series dependent on a real number x into a power series and then find the interval of convergence. \sum_{n=0}^\infty ( \sqrt{x^2+1})^n \frac{2^n }{3^n + n^3} Homework Equations After making a substitution, the book...
  27. M

    Find the power series in x-xo?

    Homework Statement Find the power series in x-x0 for the general solution of y"-y=0; x0=3. Homework Equations None. The Attempt at a Solution I'll post my work by uploading it.
  28. Elvis 123456789

    Integration by parts and approximation by power series

    Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...
  29. W

    Change a variable to transform a series into a power series

    Homework Statement The following series are not power series, but you can transform each one into a power series by a change of variable and so find out where it converges. ∑∞0 ((3n(n+1)) / (x+1)n Homework Equations a power series is a series of the form: a0 + a1x + a2x^2 ... + ... The...
  30. J

    MHB Power Series Problem: Solve f(3x) = 1/(1 - 3x)

    So here is the problem I am trying to solve: You can combine two (or more) convergent power series on the same interval I. Using the properties of the geometric series, find the power series of the function below. Series: f(x) = 1/(1 - x) = sigma k = 0, infinity = 1+ x + x^2 + x^3 Function...
  31. M

    Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0

    Homework Statement Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0. Homework Equations None. The Attempt at a Solution I'll post my whole work.
  32. T

    MHB Differentiating a power series

    I need to prove that for $-1 < x < 1$ $$\frac{1}{(1 - x)^2} = 1 + 2x + 3x^2 + 4x^3 ...$$ So, according to the textbook, the geometric series has a radius of convergence $R = 1$ (I'm not sure how this is true). In any case we can compare it to: $$\frac{1}{1 - x} =\sum_{n = 0}^{\infty} x^n$$...
  33. M

    Find the power series in x for the general solution of?

    Homework Statement Find the power series in x for the general solution of (1+x^2)y"+6xy'+6y=0. Homework Equations None. The Attempt at a Solution I got up to an+2=-an(n+3)/(n+1) for n=1, 2, 3, 4, 5, 6... a3=-2a1 a4=0 a5=3a1 a6=0 a7=-4a1 a8=0 The answer in the book says y=a0sigma from m=0 to...
  34. Houeto

    Is this ODE a Bernoulli Equation? Exploring Solutions with Substitution

    consider ODE : Show that the solution to this ODE is: Can someone tell what kind of ODE is it?I thought,it's on the form of Bernoulli ODE with P(x)=0.Is it possible to still solve it by using Bernoulli Methodology?I mean by substituting u=y^1-a with a=2? Thanks
  35. mertcan

    I Is there a proof for the precision of convergence or divergence at x=+,-(1/L)?

    hi, If you look at my attachment you can see that the book express that for the situation of x=+,-(1/L) we need further investigation. It means being converged or diverged is not precise. I would like to ask: Is there remarkable proof that if x=+,-(1/L) convergence or divergence is not...
  36. F

    How to differentiate power series starting from 2 for e^x?

    Homework Statement for d/dx(e^x) , the series should be start from 2 rather than 1 , right ? because when k = 1 , the circled part would become 0 , the series for (e^x) is 1 + x + ... Homework EquationsThe Attempt at a Solution
  37. nfcfox

    Why Is the Power Series Automatically Centered at x=2?

    Homework Statement http://imgur.com/12LbqWL Part b Homework EquationsThe Attempt at a Solution Since it says the first four terms, not nonzero, the first four terms would be 0-(1/3-0)+2/9(x-2)-1/9(x-2)^2 I'm confused when it says I need to find these for x=2... Do I just plug in x=2 now and...
  38. S

    Interval of Convergence of Power Series with Square Root

    I'm trying to find the answer to a question similar to this posted it earlier but in the wrong section I think and not explained well. $$ \sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n \ \ \rm ?$$ Find the interval of convergence? I tried using the root test and got from 0 to 1 but when I...
  39. S

    I Square Root in an alternating power series

    I had a question similar to Σ0∞ (-1)^n (x)^(n/2) and attempted to solve it using the root test getting abs(√x)<1, but I've also seen some places answer it as √abs(x)<1 so am I skipping a step.
  40. faradayscat

    Differential equation with power series

    Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...
  41. ReidMerrill

    Representing a function as a power series

    Homework Statement Represent the function (8x)/(6+x) as a power serioes f(x)=∑cnxn Find c0 c1 c2 c3 c4 Radius of convergence R= Homework EquationsThe Attempt at a Solution I've represented this function as (8x/9)∑(-x/6)n and found I-x/6I <1 so R=6 Through pure guessing I discovered c0=0 but...
  42. JulienB

    Understanding Radius of Convergence in Power Series Calculations

    Homework Statement Hi everybody! I'm a little struggling to fully understand the idea of radius of convergence of a function, can somebody help me a little? Are some examples I found in old exams at my university: Calculate the radius of convergence of the following power series: a)...
  43. Theengr7

    The decomposition of the numerator

    (a) Find a power series representation for the function. I'm struggling on the decomposition of the numerator. This exercise is from chapter 8, section 6 of Th Stewart Calculus book.
  44. nfcfox

    The power series above is the Taylor series....

    Homework Statement http://imgur.com/1aOFPI7 PART 2 Homework Equations Taylor series form The Attempt at a Solution My thought process is that the answer is 3 because using the geometric series equation (1st term)/(1-R) then you can get the sum. In this case R would be x+2 where x is -2 so 0...
  45. NihalRi

    Use geometric series to write power series representation

    Homework Statement Complete the proof that ln (1+x) equals its Maclaurin series for -1< x ≤ 1 in the following steps. Use the geometric series to write down the powe series representation for 1/ (1+x) , |x| < 1 This is the part (b) of the question where in part (a)I proved that ln (1+x)...
  46. Odious Suspect

    I Power Series solution to dy/dx=x+y

    This is from an example in Thomas's Classical Edition. The task is to find a solution to ##\frac{dy}{dx}=x+y## with the initial condition ##x=0; y=1##. He uses what he calls successive approximations. $$y_1 = 1$$ $$\frac{dy_2}{dx}=y_1+x$$ $$\frac{dy_3}{dx}=y_2+x$$ ...
  47. F

    How to Expand Noncommuting Variables in a Formal Power Series?

    Homework Statement Need to show that [a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger} Homework Equations [a,a^\dagger]=1 The Attempt at a Solution Need to expand f(a,a^\dagger) in a formal power series. However I don´t know how to do it if the variables don´t commute.
  48. Euler2718

    I Power series Construction Help

    Very basic issue here. Using: \frac{1}{1-x} = \sum_{i=0}^{\infty} x^{i} , |x|<0 Find the power series representation and interval of convergence for: f(x) = \frac{1}{(1-3x)^{2}} We have that: \frac{d}{dx}\left[\frac{1}{1-x}\right] = \frac{1}{(1-x)^{2}} = \sum_{i=0}^{\infty} ix^{i-1} ...
  49. Daniel Lobo

    Finding a limit using power series expansion

    Homework Statement The problem wants me to find the limit below using series expansion. ##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})## Homework EquationsThe Attempt at a Solution (1) For startes I'll group the two fractions inside the limit together ##\lim_{x...
  50. I

    MHB Interval of Convergence for Power Series

    Hi hi, So I worked on this problem and I know I probably made a mistake somewhere towards the end so I was hoping one of you would catch it for me. Thank you! Pasteboard — Uploaded Image Pasteboard — Uploaded Image
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