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. Amrator

    Approximating Integral via Power Series

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  2. Amrator

    Manipulating Power Series for Coefficient Extraction

    Homework Statement By considering the power series (good for |x| < 1) ##\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + x^4 +...## Describe how to manipulate this series in some way to obtain the result: ##\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2}## Homework Equations Maclaurin...
  3. J

    Why can no one explain Power Series and Functions clearly

    Hello, Im currently in a Calc II class with unfortunately a bad professor (score of 2 on RateMyProfessor), so I often have to resort to outside sources to learn. Our class is currently on Sequences and Series which has been fine up until we hit the topic of relating Power Series and Functions...
  4. H

    Power series solution, differential equation question

    I can not find a solid explanation on this anywhere, so forgive me if this has been addressed already. Given something like y''+y'-(x^2)y=1 or y''+2xy'-y=x, how do I approach solving a differential with a power series solution when the differential does not equal zero? Would I solve the left...
  5. R

    Power series where radius of convergence > lower limit

    Homework Statement Let ##\sum^{\infty}_{n=0} a_n(z-a)^n## be a real or complex power series and set ##\alpha = \limsup\limits_{n\rightarrow\infty} |a_n|^{\frac{1}{n}}##. If ##\alpha = \infty## then the convergence radius ##R=0##, else ##R## is given by ##R = \frac{1}{\alpha}##, where...
  6. D

    MHB Power Series: Find First 5 Terms of x^2/(1-5x) - Help Needed

    For this function f(x)=x^2/(1-5x). The interval of convergence is (-1/5) < x < (1/5). I tried to differentiate, but got it wrong. Could someone please help?
  7. H

    Seek power series solutions of the given differential equation

    I know there are a number of ways to do this problem, to increment the series etc. but, would someone please be able to explain how they get the answers for this problem simply and easily ? Thanks! A screen shot is attached
  8. MidgetDwarf

    Help with manipulating power series.

    So i am given (1+x)/(1-x)^2 and I have to put it into a power series. I know that 1/(1-x)= 1+x+x^2+x^3+...=∑x^n from 0 to infinity. I am having problems factoring series. I differentiate 1/(1-x). I get, 1/(1-x)^2= 1+2x+3x^2+...= ∑nX^(n-1) the sum from 1 to infinity. rewriting this equation...
  9. R

    Finding the radius of convergence of a power series

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  10. shanepitts

    Why same initial value in power series

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  11. P

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  12. nuuskur

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  13. F

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  14. Shackleford

    Find power series representations of the general solution

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  15. A

    Conceptual: Are all MacLaurin Series = to their Power Series?

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  16. J

    MHB What is the general solution for the power series in Calculus 2?

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  17. J

    Verifying a Power Series Solution for y''-4y=0

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  18. J

    Solving Power Series Problems: Finding 2 Solutions

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  19. K

    MHB Uniform convergence of a complex power series on a compact set

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  20. P

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    Homework Statement (x+1)y'' - (x-1)y' - y = 0 centred around x=1 y(1) = 2, y'(1) = 3 The Attempt at a Solution I know I am supposed to get two power series, one with a0 and one with a1 but when I am trying to figure out a pattern, I keep getting both a0 and a1 in all of my terms. So I end up...
  21. evinda

    MHB Solve Differential Equation w/ Power Series Method

    Hello! (Wave) The differential equation $y''+xy=0$ is given. Find the general solution of the differential equation (with the power series method). That's what I have tried: We are looking for a solution of the form $y(x)=\sum_{n=0}^{\infty} a_n x^n$, where the radius of convergence is...
  22. D

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  23. N

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  24. A

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  25. M

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  26. N

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  27. B

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  28. RJLiberator

    Help me decipher what this problem is asking? (Power Series)

    Homework Statement Consider the power series centered at a= 0: Σkx^(k+1) From 1 to infinity (a) Find its radius of convergence, R, and its interval of convergence. = DONE (b) For x in the interval (-R,R) find the sum of the power series. Help? Homework Equations N/a The Attempt at a...
  29. S

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  30. Abner

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  31. Julio1

    MHB Computing the Limit of a Power Series

    Compute $\displaystyle\lim_{n\to +\infty}\dfrac{1^p+2^p+3^p+\cdots +n^p}{n^{p+1}}.$
  32. L

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    I'm thinking about the following function, which is a ratio of two finite power series. I'm trying to prove the monotonicity of this function, for arbitrary K. \frac{\sum_{j=0}^k [(at)^j/j!]}{\sum_{j=0}^k [(bt)^j/j!]}, and a>b>0, t>0 I know that if k goes to infinity, the function becomes an...
  33. V

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  34. K

    Power series absolute convergence/ Taylor polynomial

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  35. M

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  36. A

    MHB Finding the Coefficient in a Power Series Sum

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  37. M

    Finding the Centre and Radius of Convergence of Power Series: Explained

    Hello. I need someone to explain to me how to find the centre and radius of convergence of power series. I got the working and the answers but there are some things I don't understand. $$\sum_{n=0}^{\infty}\frac{(4i)^n(z-i)^n}{(n+1)(n+2)}$$ Using the ratio test, we got...
  38. A

    MHB What is the centre and radius of convergence for a power series?

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  39. R

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  40. STEMucator

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  41. evinda

    MHB Can the Exponential Power Series Be Defined Without a Function?

    Hello! :cool: I am looking at the exponential power series: $$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$ It is $R=\displaystyle{\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{|a_n|}}}=\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{n!}}=+\infty$ So,the power series converges at $(-\infty,+\infty)$,so...
  42. S

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  43. B

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  44. T

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  45. S

    Find the first eight coefficients of the power series expansion.

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  46. S

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  47. A

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  48. D

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  49. B

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  50. A

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