Power series Definition and 629 Threads

Homework Statement Show that 4 = \sum from n = 1 to \infty (-2)^{n+1} (n+2)/n! by considering d/dx(x^{2}e^{-x}). Homework Equations Power series for e^{x} = \sum x^{n}/n! from 0 to \infty. The Attempt at a Solution So I started with the power series for e^{-x} = \sum -x^{n}/n...

My question is just a concept that I don't understand. When differentiating a power series that starts at n=0 we bump that bound up to n=1. My question is do we always do that? or Do we only do that when the first term of the power series is a constant and thus when it is...

Homework Statement I'm revising at the moment and a bit stumped on question 4 http://www.maths.ox.ac.uk/system/files/attachments/AC104.pdf Homework Equations The Attempt at a Solution I think for the first part of the question, the regular singular points are 0 and -2...

So we can use the Taylor's theorem to come up with a Taylor series represent certain functions. This series is a power series. So far (I'm in my second year of calc, senior in high school), I've never seen a power series that wasn't a Taylor series. So are all power series taylor series? Whether...

Homework Statement http://mathworld.wolfram.com/LegendreDifferentialEquation.html I have a question about how the website above moves from one equation to another etc. 1./ Equations (4), (5) and (6) When differentiating (4) to (5) shouldn't the the limit be from n=1, which means (5)...

[b]1. Homework Statement : Find the power series for the function f(x)=5/(7-x), centered at c=-2. [b]2. Homework Equations : a/(1-r) [b]3. The Attempt at a Solution : I know that I need to divide by seven to get (5/7)/(1-(x/7)) and then rewrite in the form the sum of (a)(r)^n. I tried adding 2...

If a_0 + (a_1)x + (a_2)x^2 + ... and b_0 + (b_1)x + (b_2)x^2 + ... are two power series and the coefficient of x^r from their product is a power series: (a_0)(b_r) + (a_1)(b_(r-1)) + ... What principle or theorem or definition(s) are we applying when finding that this is indeed the...

Homework Statement Expand e^{1/z}/\sin z in powers of z+1+i.Homework Equations Not sure, see below.The Attempt at a Solution I already know that \begin{align} \sin z & = \sum_{n=0}^\infty \frac{(-1)^{n}}{(2n+1)!}z^{2n+1} \end{align} And the other expansion for the exponential (but we just...

So this is a REALLY elementary question but I can't seem to find the answer on the net, or maybe I did but just keep skipping over it some how. (by the way, this is with respect to complex numbers z \in C which is used in Complex Analysis, thus why I chose this forum). I know what it means...

Homework Statement Represent (1+x)/(1-x) as a power series. Homework Equations The Attempt at a Solution I started with 1/ (1-x) = sum (x)^n n= 0 - infinity (1 + x) sum x^n and this is where I am stuck.Homework Statement Homework Equations The Attempt at a Solution

Hello, I'm kind of stuck in this problem. I have to express the integral as a power series. the integral of (e^x -1)/x I thought about evaluating it as f(x)=(e^x -1)/x and treating it as a Taylor series is that correct? Could I have any other hints? I would really appreciate it...

Power Series Representation of a Function when "r" is a polynomial Homework Statement Find a power series representation for the function and determine the radius of convergence. f(x)=\stackrel{(1+x)}{(1-x)^{2}} Homework Equations a series converges when |x|<1...

Homework Statement Im taking a dynamics course and I am using The strogatz book Non-linear Dynamics and Chaos I need to solve a problem that is similar to problem 6.1.14 Basically it consist in the following You have a saddle node at (Ts,Zs) which is (1,1). Consider curves passing through...

So. There's this question about power series that will eventually take the form of p= |x| lim n->inf | nn / (n+1)(n+1) | But of course, in a futile attempt at a solution I tried doing the derivative for both functions. Didn't get anywhere of course. Knowing that eventually the answer...

Homework Statement Determine the power series for g(x)=1/(1+9x)^2 The sigma in the answer has to be from n=1 to infinity We also have to specify whether it is alternating by putting either (1)^n or (-1)^n This is an online problem and I have no idea why what I am putting is not right...

Homework Statement My question involves a small algebra issue within a power series problem. I have (-1)^n-1 and i just need to know how i can re-write this. I know that if it were (1)^n+1 i could re-write as (1)^n * (n) So can i write (-1)^n-1 as (-1)^n * (-1) ? Homework Equations...

Could someone please explain the y2 solution for repeated roots in Frobenius method where y2=y1lnx+xs \Sigma CnxnI am struggling to figure out how to solve this

Homework Statement I have an infinite series that looks like this: \sum_0^\infty n(n-1)d_nx^{n-1} + \sum_0^\infty n(n-1)d_nx^n + \sum_0^\infty d_nx^nI wish to combine all three sums so that they must all have same powers of x and start at same index. The second and third summations are fine...

This is a pretty general conceptual question. I was just doing some reviewing for a test, and it occurred to me that if I were not told specifically to use Frobenius method on an equation, I might try to Power series solve it and vice versa. Can we talk about the difference a bit? We apply...

Hi, I'm new to the forum and need some help regarding my calc class. Any help you could provide would be greatly appreciated. In doing a power series series solution when should I use the frobenius method and when should I use the simple power series method. The simple method seems a little...

Homework Statement Find the power series expansion of Log z about the point z = i-2. Show that the radius of convergence of the series is R = \sqrt{5}. Homework Equations None The Attempt at a Solution I know that Log z = (z-1) - (1/2)(z-1)^2 + (1/3)(z-1)^3 -... So wouldn't this...

Homework Statement Solve x^2y'' - y = 0 using Power Series Solution expanding about xo = 2. The Attempt at a Solution First I expand the coefficient of y" (i.e. x2) about xo: TS[x^2]|_{x_o=2} = 4+ 4(x - 2) + (x - 2)^2 Assuming the solution takes the form: y(x) = \sum_0^{\infty}a_n(x -...

Homework Statement I am following along in an example problem and I am getting hung up on a step. We are seeking a power series solution of the DE: (x - 1)y'' + y' +2(x - 1)y = 0 \qquad(1) With the initial values y(4) = 5 \text{ and }y'(4) = 0. We seek the solution in the form y(x) =...

Homework Statement See figure attached. Homework Equations The Attempt at a Solution Okay I think I handled the lnx portion of the function okay(see other figure attached), but I'm having from troubles with the, \frac{1}{x^{2}} \int x^{-2} = \frac{-1}{x} + C How do I...

I'm confused between some formulae so I'm going to give some examples and you can let me know if what I'm writing is correct. Find the Taylor series for... EXAMPLE 1: f(x) = \frac{1}{1- (x)} around x = 2 Then, \frac{1}{1-(x)} = \frac{1}{3-(x+2)} = \frac{1}{3} \left( \frac{1}{1...

For the following power series, find ∑ (4^n x^n)/([log(n+1)]^(n) (a) the radius of convergence (b) the interval of convergence, discussing the endpoint convergence when the radius of convergence is finite. -------------------------------------------------------------------------------...

Homework Statement If \sum_{n=0}^{\infty} c_{n}4^n is convergent, does it follow that the following series are convergent? a) \sum_{n=0}^{\infty} c_{n}(-2)^n b) \sum_{n=0}^{\infty} c_{n}(-4)^n Homework Equations The Power Series: \sum_{n=0}^{\infty} c_{n}(x - a)^n The...

Hi, for awhile I was agonizing over part b) of this http://books.google.com/books?id=WZX4GEpxPRgC&lpg=PP1&dq=lang%20complex%20analysis&pg=PA62#v=onepage&q&f=false" of Theorem 3.2 in Lang's Complex Analysis. But I think part of the reason was that I kept concentrating on the second sentence...

Homework Statement suppose a large number of particles are bouncing back and forth between x=0 and x=1, except that at each endpoint some escape. Let r be the fraction of particles reflected, so then you can assume (1-r) is the number of particles that escape at each wall. Suppose particles...

Homework Statement In a water purification process, one-nth of the impurity is removed in the first stage. In each succeeding stage, the amount of impurity removed is one-nth of that removed in the preceding stage. Show that if n=2, the water can be made a pure as you like, but if n=3, at...

How can the tan power series be derived from the sin and cos power series? Where do the Bernoulli numbers come in?

Hi, I'm new here. I am curious that why a power series must have a radius of convergence? I mean, even in a complex plane, there is always a so-called convergent radius for a power series. Is it possible that a power series is convergent for a certain range in one direction, and for an apparent...

Homework Statement Suppose I have the power series: f(x) = A0 + A1 x +A2 x^2 ...An x^n Where A0..An are numbers, there is no recursion relation. Find the interval of convergence Homework Equations The Attempt at a Solution Can I use ratio test? How would I do this since there is no recursion...

Homework Statement http://img.photobucket.com/albums/v257/gamer567/powerseries.png Homework Equations The Attempt at a Solution http://img.photobucket.com/albums/v257/gamer567/cramster.png I am getting lost from the transition in the 1/x-2 to 1/(x-2)^2. Could someone...

Homework Statement F(x)=∫(0 to x) tan^(-1)t dt. f(x)= infinite series ∑n=1 (-1)^(en)(an)x^(pn)? en=? an=? pn=? I know en = n-1 Homework Equations The Attempt at a Solution Start with the geometric series 1/(1 - t) = ∑(n=0 to ∞) t^n. Let t = -x^2: 1/(1 + x^2) = ∑(n=0 to ∞)...

where can i find a proof of the following identity ? \sum_{n=0}^{\infty} (-x)^{n} \frac{c(n)}{n!} \sim c(x) +(1/2)c''(x)x+(1/6)c'''(x)x + (1/8)x^{2}c'''' (x) +++++

\mbox {Suppose I have: } \sum_{n=1}^\infty (\frac {x} {3})^{2n} \mbox{Can I define } y= \frac {x} {3} a_k(y) = \left\{ \begin{array}{c l} (y)^k, & \mbox{if } k= 2n\\ \\ (0)^k, & \mbox{otherwise} \end{array} \right. \mbox {And then use all the cool properties of power series on }...

I was wondering if there's an example of a power series \sum_n^\infty c_n (z-a)^n with radius of convergence R so that all z for which |z-a| = R there is purely conditional convergence? (no divergence but also no absolute convergence) Or perhaps a reason why that's impossible?

Homework Statement Find a power series expansion for log(1-z) about z = 0. Find the residue at 0 of 1/-log(1-z) by manipulation of series, residue theorem and L'Hopitals rule. Homework Equations The Attempt at a Solution Is this power series the same as the case for real numbers.

Homework Statement Let ω be the complex number e^(2πi/3)， Find the power series for e^z + e^(ωz) + e^((ω^2) z). Homework Equations The Attempt at a Solution I can show that 1+w+w^2=0, don't know if it would help. Could anyone please give me some hints? Any input is appreciated!

Hey All, I'm learning calculus through videos. The videos that I'm watching are really good, but they are deficient in power series, taylor and mclaurin series, binomial series, and taylor polynomial applications. Anyone know where I can see video instruction for these? Steve

Consider the power series (n=1 to infinity) \Sigma (x^n)/(n*3^n). (a) Find the radius of convergence for this series. (b) For which values of x does the series converge? (include the discussion of the end points). (c) If f(x) denotes the sum of the series, find f'(x) as explicitly as...

Homework Statement Find the first 6 terms of the power series expansion centered at 0 for the general solution for y -xy'=0. Then find the coefficient of the x38 term. Homework Equations General solution is of the form: y=a0+a1x+a2x2+a3x3+a4x4+a5x5+... If you factor out the ao and...

\int \frac{x-arctanx}{x^3}dx \frac{d}{dx}( x-arctanx ) = 1-\frac{1}{1+x^2}=\frac{x^2}{x^2+1} = x^2 \sum_{n=0}^{\infty}(-1)^nx^{2n} = \sum_{n=0}^{\infty}(-1)^nx^{2n+2} \int \sum_{n=0}^{\infty}(-1)^nx^{2n+2} dx = \sum_{n=0}^{\infty}(-1)^n\frac{x^{2n+3}}{2n+3}+C C=0? \int...

Homework Statement Suppose that f(z) = ∑a_j.z^j for all complex z, the sum goes from j=0 to infinity. (a) Find the power series expansion for f' (b) Where does it converge? (c) Find the power series expansion for f^2 (d) Where does it converge? (e) Suppose that f'(x)^2 + f(x)^2 = 1...

Homework Statement I need to demonstrate that \frac{\mathrm{d} }{\mathrm{d} x}\sum_{n=0}^{\infty }\frac{x^{n}}{n!}= \sum_{n=0}^{\infty }\frac{x^{n}}{n!} Homework EquationsThe Attempt at a Solution I just need a hint on how to start this problem, so how would you guys start this problem?

Homework Statement Homework Equations We're using generating functions, and recurrence relations of homogeneous and non-homogeneous types The mark allocation is 2, 3, 3 and 2 The Attempt at a Solution I think I've done the first part correctly. The closed form is in terms...

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. From classical mechanics, TcosX= W and TsinX=F. Find F/W as a power series of X(angle). Often in a...

title is pretty much the jist of it.

hi! are the following power series equivalent? ln(1+x)=\sum_{n=0}^{\infty} \frac{(-1)^n n! x^{n+1}}{(n+1)!} =\sum_{n=0}^{\infty} \frac{(-1)^n x^{n+1}}{n+1}