Power series Definition and 629 Threads

Given a power series, what is the condition on its coefficients that means the ratio test can be applied?

Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x) = arctan(x/3) Homework Equations not really any equations ... just intervals The Attempt at a Solution here's what i did: f'(x) = \frac{1}{1+(x/3)^2} =...

Homework Statement Find the radius of convergence and interval of convergence of the series: \sum (-1)n * ( x^(2n) / (2n)! ) for n=0 to infinity Homework Equations well, any of the divergence tests: ratio, limit comparison, etc. The Attempt at a Solution I don't get power...

I'm trying to prove eA eB = eA + B using the power series expansion eXt = \sum_{n=0}^{\infty}Xntn/n! and so eA eB = \sum_{n=0}^{\infty}An/n! \sum_{n=0}^{\infty}Bn/n! I think the binomial theorem is the way to go: (x + y)n = \displaystyle \binom{n}{k} xn - k yk = \displaystyle...

It just occurred to me now that I never asked why the Maclaurin series of $e^u$, u being some function of x, is simply $\sum\frac{u^n}{n!}$. I should say I understand why the Maclaurin series of $e^x$ is $\sum\frac{x^n}{n!}$, due to the derivative of the exponential function being itself, but...

Homework Statement If f(z) = \sum an(z-z0)n has radius of convergence R > 0 and if f(z) = 0 for all z, |z - z0| < r ≤ R, show that a0 = a1 = ... = 0. Homework Equations The Attempt at a Solution I know it is a power series and because R is positive I know it converges. And if...

I'm trying to find the power series expansion of 1/1+z at z=-5 and the radius of convergence.How should I think and solve this problem? I'm looking for a step by step explanation because I want to understand the mechanics behind it.Thank you.

$\displaystyle f(z) = \frac{4 + 3z}{(z + 1)(z + 2)^2}$ How do I find the power series? I know that $\displaystyle\frac{1}{z+1} = \frac{1}{1-(-z)} = \sum_{n}^{\infty}(-z)^n$ and $\displaystyle\frac{1}{(z+2)^2} = \frac{d}{dz} \frac{-1}{z+2} = \frac{d}{dz} \frac{-1}{1 - (-z-1)} =...

z is a complex number. f(z) = \frac{4 + 3z}{(z + 1)(z + 2)^2} \frac{1}{1 + z} = \frac{1}{1 - (-z)} = 1 + (-z) + (-z)^2 + \cdots = \sum_{n = 0}^{\infty}(-z)^n \frac{1}{(z + 2)^2} = -\frac{d}{dz} \ \frac{1}{1 - (-z - 1)} = -\frac{d}{dz}\sum_{n = 0}^{\infty}(-z - 1)^n = \sum_{n =...

z\in\mathbb{Z} \frac{1}{1-(-z)}=\sum_{n=0}^{\infty}(-z)^n \frac{1}{(z+2)^2}=\frac{d}{dz} \frac{-1}{1-(1-z)} = \frac{d}{dz} (1 + (1-z) + (1-z)^2+\cdots = 0 -1 -2(1-z)-3(1-z)^2 - \cdots = \sum_{n=0}^{\infty} ? Not to sure about the second one.

Homework Statement We know that y = Aex is the solution to the initial value problem dy/dx = y; y(0) = A. This can be shown by solving the equation directly. The goal of this problem is to reach the same conclusion using power series. Method: Let y be a solution to the initial value...

Hello, In my QM class we're using power series which don't converge but apparently still give a good approximation if one only takes the lower-order terms. Is there any way to understand such a phenomenon? Is it a genuine area of mathematics? Or is it impossible to say something general on...

Homework Statement My professor says that the integral test is supposed to help me figure out the interval of convergence of this equation: Ʃ(2n*(x+1)n)/(n*ln(n)) Homework Equations integral test The Attempt at a Solution i couldn't figure out what the power series was after i...

Suppose I have the Laurent series with region of convergence given below: f(z)=\sum_{n=-\infty}^{\infty} a_n z^n,\quad \sqrt{3}<|z|<\sqrt{5} Can I conclude the Laurent-Puiseux series: f(\sqrt{z})=\sum_{n=-\infty}^{\infty} a_n \left(\sqrt{z}\right)^n has a region of convergence...

Homework Statement Expand f(x)= (x+x2)/(1-x)3 Homework Equations ? The Attempt at a Solution I've tried everything I can think of to simplify this equation: substitution of various other power series, partial fraction decomposition, taking derivatives, multiplying out the...

I have to find the sum of the power series: \sum_{n=1}^\infty nx^{n+1} I know the I'm supposed to show work but I don't have any idea where to start. I'm not asking for you to do the problem for me, just a hint. The only idea I had was to take the derivative to get rid of the n+1 in the...

Write a power series representation of xln(1+x2) My first instinct was to attempt to take the second derivative and then find the summation and then integrate but that approach seemed to be a dead end. Basically, the x thrown in there confuses me and you can't split the function into two...

Power Series:(n^n)*(x^n) Homework Statement The only step I'm having problem with on this problem is when I take the Lim n→∞ of the problem. I want to know how to cancel the n^n on the denominator during one of the steps. Homework Equations ∞ Ʃ (n^n)*(x^n) n=1 The Attempt at a...

f(x) = \sum_{n=1}^{\infty} \frac{(-1)^nx^{n}}{(n)^{\frac{3}{2}}} Prove f(x) is strictly monotonic (where f is defined) and that there exists one solution to f(x)=1.5 and f(x)=-0.5 First, how do I show that the radius of convergence is between -1 \le x \le 1? And then, how do I...

what does it mean if y(x)=\sum^{∞}_{n=0} an(x-x0)n and y(0)=1 ? does this mean that x0=0 and a0=1 ?

Find y(x) as a power series satisfying: y'-2xy=0 , y(0) = 1 attempt: y=\sum^{∞}_{n=0} an(x-x0)n y'=\sum^{∞}_{n=0} (n+1)an+1(x-x0)n y' - 2xy = y' - 2y(x-x0) - 2x0y substituting the power series into above formula and simplifying eventually gives: a1-2x0a0 +...

Homework Statement The behavior of a neutron losing energy by colliding elastically with nuclei of mass A is described by a parameter ξ1, ξ1 = 1 + \frac{(A-1)^2}{2A}*ln\frac{A-1}{A+1} An approximation, good for large A, is ξ2= \frac{2}{A+2/3} Expand ξ1 and ξ2 in powers of A−1. Show...

Homework Statement I have a problem set that asks me to determine, first, the radius of convergence of a complex series (using the limit of the coefficients), and second, whether or not the series converges anywhere on the radius of convergence. Homework Equations As an example: Σ(z+3)k2 with...

Homework Statement "Derive a more accurate approximation for the heat capacity at high temperatures, by keeping terms through x^{3} in the expansions of the exponentials and then carefully expanding the denominator and multiplying everything out. Throw away terms that will be smaller than...

Hello, I've been working on solving the equation y''-2xy'+2py=0. where p is a positive integer. I've assumed y=\sum a_{n}x^{n} for n=0 to inf I'm getting two formulas for a_{n} One is for odd n, the other for even n, related to a_{0} and a_{1} However, the relation involves something that...

Homework Statement i was doing this exercise and came across this example. ∞ Ʃ (x^n)/ln(n+1) n=1 The Attempt at a Solution i know you have to do the ratio test which is lim | a(n+1)/a(n)| n>∞ i got to lim | [x ln(n+1)] /ln(n+2) | n>∞ and have no idea how to continue? is...

I've just started learning to solve DE by using power series and I am not sure If I did it the right way. Would anybody be kind enough going thru my solution here and see if I did it right? I want to make sure I am doing the right thing, I am very bad with series. Thanks in advance!and i have...

Homework Statement Hi there, I have just started taylor series for my course.. most seems arlgiht so far, however when it comes to validating a given series( tayor or maclaruin), I have an idea on how to find out the x value.. but I don't know what I am doing wrong.Take for example: The...

Hi Every body! I wan to compute the power series expansion of dedekind eta function. Specifically, I want to know the power series expansion of η(τ)/η(3τ)? How could I expand this function? I would be happy if you could help me as I am stuck at this state when I am computing the modular...

Homework Statement I am asked to comment on the convergence/divergence of three series based on some given information about a power series: \sum_{n=0}^{\infty}c_nx^n converges at x=-4 and diverges x=6. I won't ask for help on all of the series, so here's the first one...

Homework Statement Find the power series representation for the function f(x)=x/(x^2-3x+2) and determine the interval of convergence. Homework Equations The Attempt at a Solution First I separate into partial fractions 2/(x-2) - 1/(x-1) 2/(x-2) = sum n=0 to infinity (x/2)^n...

Hey guys, sorry for sending out so many questions so fast. I just discovered this site, and it looks great. Plus, I have my first complex analysis midterm tomorrow, so I'm pretty stressed (you'd think after 4 years of math/econ/computer science you'd get used to it but there's nothing like the...

I want to show that e^x e^x = e^{2x} using a power series expansion. So I start with \sum_{n=0}^\infty \frac{x^n}{n!} \sum_{m=0}^\infty \frac{x^m}{m!} \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^n}{n!} \frac{x^m}{m!} \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^{m+n}}{m!n!} I am...

I having trouble trying to compute the sum of a finite power series with matlab.Basically, the problem says to take x=0.5 and compute the sum of a series for n=5, n=10, n=25, and n=100 and for all coefficients ai=1. It also says to use the operation .^ and command sum(v) that takes a vector and...

Homework Statement Using power series, find the solution to the following DE y" + xy' + y=0 Given data xo = 0 y(0)=1 and y'(0) = 1 Homework Equations y(x) = \suman(x-xo)^n for n= 0 to \infty The Attempt at a Solution Please see attached file. The part circled at the...

Homework Statement Using power series, find solutions to the following DE y' + y= x^2, y(1)= 2 and xo=1 Homework Equations y(x)=an\sum(x-xo)^n for n=1 to infinity The Attempt at a Solution See the attachment NOTE: I only want to find a way to collect all the x...

Homework Statement Find the radius of convergence and the interval of convergence of the series sigma[n=1,inf] ((3x-2)^n/(n^2*3^n)) Homework Equations The Attempt at a Solution sigma[n=1,inf] ((3x-2)^n/(n^2*3^n)) I applied the Root Test p=lim n->inf |(3x-2)^n/(n^2*3^n)|^(1/n) = lim n->inf...

Homework Statement A heavy weight is suspended by a cable and pulled to one side by a force F. How much force is required to hold the weight in equilibrium at a given distance x to one side. Tcosθ=W and Tsinθ=F. Find F/W as a power series of θ. Often in a problem like this, what we know is not...

Homework Statement Find first three non zero terms in series expansion for ln(1+e^-z) when z is very large Homework Equations The Attempt at a Solution I've got as far as ln(1+e^(-1/z)*e^((1/z)(z^(2) - 1)) not sure where to go from here

Homework Statement Perform a partial fractions expansion of 1/(n(n+1))=a/n + b/(n+1) in order to find a function that represents x^n/(n(n+1)). Homework Equations The Attempt at a Solution so i broke up the partial fraction to 1/n -1(1+n). I integrate both to get ln(n) +...

Homework Statement (1+2x^2)y''+6xy'+2y=0 1. find the power series solutions of the equation near x0=0...show the recurrence relation for an, derive a formula for an in terms of a0 and a1, and show the solution in the form y=a0y1(x)+a1y2(x) 2.what is the lower bound for the radii convergence...

Homework Statement Does there exist a power series expansion of log z around z=0? If so, what is it? If not, demonstrate that it is impossible. Homework Equations [PLAIN]http://img839.imageshack.us/img839/4839/eq1.gif Here is the expansion of log z around z=1. The Attempt at a Solution...

Homework Statement Solve the fluxional equation (y with a dot on top)/(x with a dot on top) = 2/x + 3 - x^2 by first replacing x by (x + 1) and then using power series techniques.Homework Equations dy/dx = 2/x + 3 - x^2 The Attempt at a Solution First, I believe the fluxional (y with a dot...

I know how to do problems like y' + y = 0 where you can replace y' and y with a series in sigma notation, manipulate and compare coefficients. But how do you solve a differential by power series that does not also include y or a higher order derivative? For example, y' = -(x^2) + 2/x + 3...

Homework Statement a) Determine the series of the given function. In the first box after the summation symbol, type in -1 or 1 indicating whether the series is alternating or not. b) Write out the sum of the first four nonzero terms of the series representing this function. c) Determine the...

a) Determine the series of the given function. In the first box after the summation symbol, type in -1 or 1 indicating whether the series is alternating or not. b) Write out the sum of the first four nonzero terms of the series representing this function. c) Determine the interval of...

Homework Statement Find the particular solution to the ODE y"+y=x using power series Homework Equations y=\sum(a_{n}x^{n})The Attempt at a Solution i tried plugging in y=\sum(a_{n}x^{n}) into the original equation and comparing coefficients of x to the first degree, but i am not sure how to...

Homework Statement sketch on the complex plane the region where the following two power series both converge 1) sigma from n=0 to infinity [(z-1)^n]/[n^2] 2) sigma from n=0 to infinity [((n!)^2)((z+4i)^n)]/[2n]! The Attempt at a Solution R=lim as n tends to infinity...

So I've gotten into the Method of Frobenius and all; Solved a few questions, however the most inconvenient part would be the formulation of the general equations for the final answer. Granted, the lecturer told us to not spend so much time on that segment due to its minimal weightage, but I...

Homework Statement consider the initial value problem (1-x)y,,+xy,-2y=0 find the series solution up to the term with x6 Homework Equations (1-x)y,,+xy,-2y=0 The Attempt at a Solution assuming the answer has the form \Sigmaanxn that gives y,,=\Sigmananxn-1 and...