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. ognik

    MHB Can Series Expansion Prove the Relation Between Inverse Coth and ln(x+1)/(x-1)?

    Hi - my sometimes surprising set-book asks to show by series expansion, that $ \frac{1}{2}ln\frac{x+1}{x-1} =coth^{-1} (x) $ I get LHS = $ x+\frac{{x}^{3}}{3}+\frac{{x}^{5}}{5}+... $, which I think $= tanh^{-1} $ but I have found different expansions for the hyperbolic inverses, so I'd...
  2. Amrator

    Approximating Integral via Power Series

    Homework Statement Approximate the integral to 3 decimal place accuracy via power series. ##\int_0^{1/2} x^2 e^{-x^2}\, dx ## Homework EquationsThe Attempt at a Solution ##x^2 e^{-x^2} = x^2 \sum_{n=0}^\infty \frac {(-x)^{2n}}{n!} = \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}## ⇒ ##\int_0^{1/2}...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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?
  8. 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
  9. 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...
  10. R

    Finding the radius of convergence of a power series

    Homework Statement Σ(n=0 to ∞) ((20)(-1)^n(x^(3n))/8^(n+1) Homework Equations Ratio test for Power Series: ρ=lim(n->∞) a_(n+1)/a_n The Attempt at a Solution I tried the ratio test for Power Series and it went like this: ρ=lim(n->∞) (|x|^(3n+1)*8^(n+1))/(|x|^(3n)*8^(n+2)) =20|x|/8 lim(n->∞)...
  11. shanepitts

    Why same initial value in power series

    Homework Statement Find a power series representation for the function below & determine the radius of convergence. f (x)=(1+x)/(1-x)2 2.Relevant equation Shown in attached image below which is the solution the problem. 3.The attempt at a solution I'm trying to fathom the solution here...
  12. P

    Question regarding Power Series

    Homework Statement It is stated in my textbook that the sum ## \sum_{0}^{\infty} 8^{-n}(x^2-1)^n ## is not a power series but can be turned into one using he substitution ##y=x^2-1## which then becomes the power series ##\sum_{0}^{\infty} 8^{-n}y^n ## They aren't offering any explanation as to...
  13. nuuskur

    Interval of convergence and sum of power series

    Homework Statement Consider ##\sum\limits_{n=0}^{\infty} \frac{n+1}{(2n)!}(x+1)^{2n+1}##. Find the interval of convergence and sum of the power series. Homework EquationsThe Attempt at a Solution According to the textbook: given the power series ##\sum a_n(x-c)^n## the radius of convergence...
  14. F

    E^A matrix power series (eigen values, diagonalizable)

    Homework Statement Find an expression for e^A with the powerseries shown in the image linked Homework Equations I know you have to use eigen values and eigen vectors and a diagonal matrix The Attempt at a Solution What I did was just try to actually multiply out the infinite series given. I...
  15. Shackleford

    Find power series representations of the general solution

    Homework Statement (1+x2) y'' + 2xy' = 0 in powers of x Homework Equations y'' = \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} y' = \sum_{n=1}^{\infty} na_nx^{n-1} The Attempt at a Solution (1+x2) y'' + 2xy' = (1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=1}^{\infty} na_nx^{n-1}...
  16. A

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

    Homework Statement To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...
  17. J

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

    please help!
  18. J

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

    Homework Statement substitute the given power series below into ODE y'' -4y=0 to verify it is a solution Homework Equations y=∑ 2n xn / n! n=0 y''-4y=0 The Attempt at a Solution I have absolutely no idea how start.
  19. J

    Solving Power Series Problems: Finding 2 Solutions

    Hello. I've been solving power series problems where the question asks to find 2 power series solutions. I can solve it almost all, I can find the recurrence solution... however while checking the solutions, I see some answer solutions they used c0=1 and c1=0 to find the 2 solutions, and some...
  20. K

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

    I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...
  21. P

    Power series solution to degree 2 ODE

    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...
  22. 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...
  23. D

    Slight confusion in proof of Hadamard's Lemma

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  24. N

    Complex Analysis: Special Power Series

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  25. A

    MHB Ideals of formal power series ring

    I need help understanding the following solution for the given problem. The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i \in F$ forms a ring $F[[t]]$. Determine the ideals of the ring. The solution: Let $I$ be an...
  26. M

    De Moivre's Theorem and Power Series

    Homework Statement Hi I'm stuck with the following question: Use de Moivre's Theorem and your knowledge of power series to show: 1/1(1/2^1)cos(θ)+1/2(1/2^2)cos(2θ)+1/3(1/2^3)cos(3θ)+ ... = log(2)-1/2*log(5-4cos(θ))Homework EquationsThe Attempt at a Solution I have already established the...
  27. N

    Using Power Series to Evaluate ln and sin at a Given Point

    Homework Statement "Use power series to evaluate the function at the given point" ## ln (x+ \sqrt{1+{x^2}}) - sin x ## at ## x = 0.001 ## Homework Equations Relevant power series: A: ## ln (1+x) = \Big( \sum_{n=0}^\infty\frac{({(-1)^{n+1}}{x^n})}{n} \Big) ## B: ## {(1+x)^p} =...
  28. B

    Power series of a strange function

    1. Write ∫e^(-t^2)dt with 0<=t<=x , as power series around 0. For what values of x this series converge ? attempt at a solution: f' = e^(-x^2) => f'(0) = 1 f''= -2x*e^(-x^2) => f''(0)= 0 f'''= -2e^(−x2) +4*x^2*e^(−x^2) => f'''(0)=-2 I tried to find a general rule for the derivatives but with...
  29. 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...
  30. S

    Can someone help with my differential equation involving power series?

    Homework Statement Find two power series solutions of the differential equation about the ordinary point x = 0. Homework Equations y'' + x^2y' +xy = 0 The Attempt at a Solution Check attachment. I found my y1 and y2, the boxed in answers are the ones the book says are the answers. Can...
  31. Abner

    Representing functions as power series

    Homework Statement I have this function f(x) = \frac{6}{1+49x^2}, and i suppose to represent this function as a power series \displaystyle f(x) = \sum_{n=0}^\infty c_n x^n. Then i need to find the first few coefficients in the power series. Homework EquationsThe Attempt at a Solution After...
  32. Julio1

    MHB Computing the Limit of a Power Series

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

    Monotonicity of the ratio of two power series

    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...
  34. V

    Power series solution to differential equation

    Homework Statement Find the power series solution of the differential equation y''-\frac{2}{(1-x)^2}y=0 around the point ##x=0##. Homework Equations y=\sum_{n=0}^\infty{}c_nx^n y'=\sum_{n=0}^\infty{}c_{n+1}(n+1)x^n y''=\sum_{n=0}^\infty{}c_{n+2}(n+2)(n+1)x^n The Attempt...
  35. K

    Power series absolute convergence/ Taylor polynomial

    1. What if absolute convergence test gives the result of 'inconclusive' for a given power series? We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...
  36. M

    Finding f^n(0) for the Power Series e^(αLn(z+1))

    Hi. I have another question about power series. I am having problem with the summarizing of the sum (writing in $\sum_{}^{}$ form). Here is the question: Let α be a real number that is not 0. Let $$f(z)=e^{{\alpha}Ln(z+1)}$$ For integer n>0, find $$f^n(0).$$ My partial solution...
  37. A

    MHB Finding the Coefficient in a Power Series Sum

    Hi. I have another question about power series. I am having problem with the summarizing of the sum (writing in $\sum_{}^{}$ form). Here is the question: Let $\alpha$ be a real number that is not 0. Let $f(z)=e^{{\alpha}Ln(z+1)}$ For integer n>0, find $f^n(0)$. My partial solution...
  38. 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...
  39. A

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

    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 $$\lim_{{n}\to{\infty}}...
  40. R

    Power series and Laplace transform

    As it can be read here, http://en.wikipedia.org/wiki/Laplace_transform#Relation_to_power_series the Laplace transform is a continuous analog of a power series in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by exp(-s). Therefore, computing a...
  41. STEMucator

    Can Power Series Expansions be Centered About Any Point?

    I had a quick question about an expansion. Wolfram and maple have not been very useful in verifying the series. Could I do these: Centered about a=1: ##f(x) = e^{-3x^2} = e^{-3}e^{-3(x^2-1)} = \sum_{n=0}^{∞} \frac{e^{-3}(-3)^n}{n!} (x^2-1)^n## Centered about a=2: ##f(x) =...
  42. 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...
  43. S

    Find power series if you know its laplace transformation

    Homework Statement a) Determine power series ##\sum _{n=0}^{\infty }a_nt^n## if you know that its laplace transformation is ##-s^{-1}e^{-s^{-1}}## b) Determine function ##g## that this power series will be equal to ##J_0(g(t))##Homework Equations The Attempt at a Solution Hmmm, I am having...
  44. B

    Solve differential equation using power series

    Homework Statement Solve ##y^{''}+zy=0## where ##y(0)=0## and ##y^{'}(0)=1##Homework Equations ##y(z)=z^r\sum _{k=0}^{\infty } C_kz^k## The Attempt at a Solution Well firstly: ##r(r-1)+p_0r+q_0=0## where obviously ##p_0=q_0=0## so ##r_1=0## and ##r_2=1##. In general ##y(z)=\sum...
  45. T

    Sum of a Power Series: Finding the Sum of a Series with a Variable

    Homework Statement find the sum of the following series: \sum_{n=1}^\infty nx^{n-1} , |x|<1 Homework Equations \frac{a}{1-r} The Attempt at a Solution i know that a function representation for that series is -\frac{1}{(1-x)^2} but how is it possible to find the sum of a series with a...
  46. S

    Find the first eight coefficients of the power series expansion.

    Homework Statement Problem: Find the first eight coefficients (i.e. a_0, a_1, a_2, ..., a_7) of the power series expansion y = ##Σ_{n = 0}^{∞}## [##a_n## ##x^n##] of the solution to the differential equation y'' + xy' + y = 0 subject to the initial-value conditions y(0) = 0, y'(0)...
  47. S

    Power Series Help: Find Interval of Convergence

    Homework Statement Find interval of convergence and radius of convergence of the following infinite series. Series from n=1 to infinity ((-3)^n * x^n) / (n*(n)^1/2)Homework Equations Ratio testThe Attempt at a Solution I've started with the ratio test and end up getting 3xn^(3/2) /...
  48. A

    MHB Solving Power Series for 9/25: Find x When y = 9/25

    I was asked to find sums equal to 9/25 by using the power series of y=\frac{1}{1+x^2}. First thing I did was to find the power series representation of the function: \sum_{n=0}^{\infty }(-x^2)^n Next I figured out the interval of convergence: \left \| -x^2 \right \|< 1 This meant that x...
  49. D

    Power series (expansion series)

    Homework Statement hi, for the expansion of power series (logarithmitic series) , ln(1+X) , why the condition for x is between -1 and 1 which x can be 1 but x can't be -1 ? Homework Equations The Attempt at a Solution
  50. B

    Power Series Solution of Differential Equation

    Homework Statement (x^2)y' = y Homework Equations The Attempt at a Solution Plugging in series everywhere I get the equation \sum na_{n}x^{n+1} = \sum a_{n}x^{n}. I try to set the coefficients for the corresponding powers equal, but when I do I don't get the correct answer. I also...
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