# 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. ### Interval of convergence of power series from ratio test

If ##a_n\neq 0## for all ##n##, consider the limit $$\lim\limits_{n\to\infty} \left | \frac{a_{n+1}x^{n+1}}{a_nx^n} \right | = \lim\limits_{n\to\infty}\left |\frac{a_{n+1}}{a_n}\right | |x|=L$$ The ratio test asserts that this series converges if ##L<1## and diverges if ##L>1##. I'm a bit...
2. ### I Differential equation using power series method

I am attempting to solve this differential equation with power series I came with the following solution but I doubt it is correct. Since x=1 we get: I doubt its correctness because it looks messy. Also the convergence radian R goes to 0, giving only a solution for x=0 which is not...
3. ### I On a remark regarding the Cauchy integral formula

The way the formula is stated in my noname lecture notes is as follows: Then they remark that: The last sentence puzzles me deeply, specifically the part "When ##|z_0|<R##...". Why can we expand ##\frac1{z-z_0}## when ##|z_0|<R##? This makes little sense to me. I have noticed several typos...
4. ### Power series product convergence

I take $$P(z)=\frac {1-z}{5-z} = \frac 1 5 -\frac 4 {25} z - \frac 4 {125} z^2 - \cdots$$ which has radius of convergence 5, and $$Q(z)=\frac {5-z} {1-z} = 5+4z+4z^2+\cdots$$ which has radius of convergence 1. ##P(z)Q(z)=1## converges everywhere. Is this correct? If so, do you think it's a good...
5. ### I On (real) entire functions and the identity theorem

In Ordinary Differential Equations by Adkins and Davidson, in a chapter on the Laplace transform (specifically, in a section where they discuss the linear space ##\mathcal{E}_{q(s)}## of input functions that have Laplace transforms that can be expressed as proper rational functions with a fixed...

35. ### Differential equation with power series method

Homework Statement I need to solve the DE y’ = x^2y using the power series method Homework Equations y = sum(0->inf)Cnx^n y’ = sum(1->inf)nCnx^(n-1) The Attempt at a Solution I plug in the previous two equations into the DE. What is the general procedure for these problems after that...
36. ### MHB Power Series .... Abbott, Theorem 6.5.1 .... ....

I am reading Stephen Abbott's book: "Understanding Analysis" (Second Edition) ... I am focused on Chapter 6: Sequences and Series of Functions ... and in particular on power series ... I need some help to understand Theorem 6.5.1 ... specifically, some remarks that Abbott makes after the proof...
37. ### Find the expression for the sum of this power series

Homework Statement Hello, I need to find an expression for the sum of the given power series The Attempt at a Solution I think that one has to use a known Maclaurin series, for example the series of e^x. I know that I can rewrite , which makes the expression even more similar to the...
38. ### MHB Power Series (Which test can i use to determine divergence at the end points)

Hello, I was given f(-4x)= 1/(1+4x), and I used the geometric series to find the power series representation of this function. I then took the limit of (-4x)^k by using ratio test. The answer is abs. value of x. So -1/4<x<1/4 I then plugged in those end points to the series going from k=0 to...
39. ### 1st Order Differential Equation - Power Series Method

Homework Statement The Attempt at a Solution I have deliberately made several obvious steps, because I keep ending up here. However I have no idea what to do from here. I thought about the fact, that differential equations have the solution ##x = x_{HOM} + x_{Inhom}##, but the ##x_{HOM}##...
40. ### MHB Calculus 2 (Power Series) when the limit is zero by root test

Hi guys! Here's a problem i was working on. I solved it by root test and got absolute value of x on the outside of the limit and the limit equaled zero. Is it wrong to multiply the outside absolute value by the zero I got from the limit? or is that okay? In general, when we are solving power...
41. ### I Y'' + y = 0 solution and recursion relation

I've found the general solution to be y(x) = C1cos(x) + C2sin(x). I've also found a recursion relation for the equation to be: An+2 = -An / (n+2)(n+1) I now need to show that this recursion relation is equivalent to the general solution. How do I go about doing this? Any help would be...
42. ### Radius of Convergence and Power Series Expansion Around Multiple Values

Homework Statement Suppose we are given the power series expansion ##f(x) = -\sum_{n=1}^{\infty} \frac{x^{n}}{n} ## which converges for |z|<1. What is the radius of convergence? Sum this serie and derive a power series expansion for the resulting function around -1/2, 1/2, 3/4 and 2. The...
43. ### Simplification step: solving a diff eqn using a power series

Homework Statement Hello, I suspect this is an easy answer but I am not seeing it. I am reviewing (more so for fun / hobby) some differential equations – I’m not in school. I’m needing help with an example problem in Differential Equations With Boundary-Value Problems Zill 2nd edition. In...
44. ### Finding a the value of 30th derivative given power series.

Homework Statement The problem is attached as pic Homework Equations ∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
45. ### Radius of convergence of the power series (2x)^n/n

Homework Statement in title Homework EquationsThe Attempt at a Solution so i know that i have to use the ratio test but i just got completely stuck ((2x)n+1/(n+1)) / ((2x)n) / n ) ((2x)n+1 * n) / ((2x)n) * ( n+1) ) ((2x)n*(n)) / ((2x)1) * (n+1) ) now i take the limit at inf? i am stuck here i...
46. ### Power Series and Convergence for ln(x+1)

Homework Statement What is the power series for the function ln (x+1)? How do you find the sum of an infinite power series? Homework Equations sigma from n=1 to infinity (-1)^n+1 (1/n2^n) That is the power series, how is that equivalent to ln (x+1)? How do you find the sum, or what does it...