Power series Definition and 644 Threads

1. Express the function as the sum of a power series by first using partial fractions. Find the interval of convergence. f(x)= (7x-1)/(3x^2 +2x-1) 2. using the fact that 1/(1-x) =\sum from \infty to n=0 of x^n and the interval for convergence of that is (-1,1) 3. I know that...

Homework Statement Find the radius and interval of convergence for the following power series. \sum_{n = 2}^{\infty}\frac {(1 + 2cos\frac {\pi n}{4})^n}{lnn}x^n The Attempt at a Solution R = \frac {1}{\lim_{n\rightarrow\infty}\sqrt [n]{\frac {(1 + 2cos\frac {\pi n}{4})^n}{lnn}}}...

I have learned that if a function of one real variable can be defined as a power series, then this one is its Taylor series. Does the same occur with functions of 2 real variables? I mean, if a function f(x, y) can be defined as a power series, does this series is the Taylor series of f(x...

Homework Statement The function f(x)=10xarctan(5x) is represented as a power series http://img464.imageshack.us/img464/4131/formub5.jpg Find the first few(5) coefficients in the power series. Homework Equations I already know that the representation of arctanx is summation from...

For the fun of it, my DE book threw in a couple of problems involving nonhomogenous second order DE's in the section I'm currently going through. Although I have solved for the complementary solution, any suggestions on how to find the particular solution? For example, the one I'm looking at...

how could i expand something such as arctan'x into a power series. also how would you be able to find the power series for it?so far i have managed to work out that: arctan'x = \frac{1}{1 + x^2} \frac{1}{1+x^2} = 1 - x^2 + x^4 - x^6 +...+ (- 1)^n x^{2n} how do you work out the radius of...

1. I this from my homework solution. (1-t/s)^n = exp(-t/s) as n goes to infinity I don't understand. I checked the exponential power series. It should be : exp(x) = summation (x^n / n!) n=0 to infinity How come it could be a exponential function ? 2. another is that...

y''+(x^2)y = 0 I tried to solve this problem using Power Series.But i can't make the solution in the form of series that have only two constants(a0,a1)that is, there are a0,a1, a2, a3. So i just wonder how can i make it has two constants.

Hi I stumbled across a power series pattern while working on a C algorithm and was wondering if this "discovery" has a name/reference anywhere. Basically, Here's what I found: for p = 1 the minimum number of terms, on the left side, satisfying the following is 1 a^p + b^p + c^p ... =...

I'm trying to do the question attached. I got the first three answers correct knowing that the nth derivative of a function evaluated at 0 divided by n! = c_n. However, I did the same for the others and the answer is incorrect. I know that I need the power series representation of that function...

My exam is coming up, I have 2 questions on infinite series. Any help is appreciated!:smile: Quesetion 1) http://www.geocities.com/asdfasdf23135/calexam1.JPG For part a, I got: g(x)= Sigma (n=0, infinity) [(-1)^n * x^(2n)] For part b, I got: x ∫ tan^-1...

Homework Statement Start with the power series representation 1/(1-x) = sum from n=0 to inf. of x^n for abs(x) < 1 to find a power series representation for f(x) and determine the radius of convergence. f(x)=ln(5+x^2) Homework Equations The Attempt at a Solution Okay, so I...

Homework Statement Find a power series representation for the function f(x) = x / (4+x) and determine the interval of convergence. I have no idea how to begin this problem. My only guess would be trying to divide something out in order to simplify to something that I'm able to...

Power series and integration >:( Hello, this question is about power series and integration of power series, the question and my working is on the image below. I had to write the question and my working plus the correct answer on a piece of paper and scan it, sorry for the hasle...

There's a homework problem that I've been struggling over: Find a formula for the truncation error if we use 1 + x^2 + x^4 +x^6 to approximate 1/(1-x^2) over the interval (-1, 1). Now, I assume that you need to use LaGrange error but I'm not sure how to proceed. Any help would be greatly...

If f(x) has a power series: a_n(x-a)^n (centered at a) what does the power series for f(2x) look like?

Homework Statement Show that the power series \sum_{k=1}^{k=\infty} \frac{x^{2k+1}}{k(2k+1)} converges uniformly when |x| \leq 1and determine the sum (at least when |x| < 1). The Attempt at a Solution Couldn't I somehow go about and show that, as |x| \leq 1, then f =...

How do I multiply power series? Homework Statement Find the power series: e^x arctan(x) Homework Equations e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} arctan(x) = 0 + x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} The Attempt at a Solution So do I multiply 1 by 0, x by...

I've tried solving the equation xy'(x) - y(x) = x^2 Exp[x] using the power series method. I assume that y has the form: y = \sum_{n=0}^{\infty} a_n x^n Inserting this in the diff. eq. gives: \sum_{n=0}^{\infty} n a_n x^n - \sum_{n=0}^{\infty} a_n x^n = x^2 e^x Now, in the other...

Homework Statement Find the radius of convergence of the following series. \sum\limits_{k = 1}^\infty {2^k z^{k!} } Homework Equations The answer is given as R = 1 and the suggested method is to use the Cauchy-Hadamard criterion; R = \frac{1}{L},L = \lim \sup \left\{ {\left|...

when finding a power series solution we have to put the differential equation ay''+by'+c=0 into the form y''+By+C=0 this leads to singular points when a=0 but why can't we leave the equation in its original form and use power series substitution to avoid singular points? or in...

Homework Statement I have f(x) = the series x^k/[(k-1)k] summed from x=2 to infinity, and I need to find its closed form. Hint: What is the derivative of f(x) Homework Equations None. The Attempt at a Solution To start this problem, I took the derivative of the series, which...

Homework Statement find a power series representation for the function and determine the radius of convergence. f(t)= ln(2-t) Homework Equations The Attempt at a Solution i first took the derivative of ln(2-t) which is 1/(t-2) then i tried to write the integral 1/(t-2)...

Homework Statement "Find the radius of convergence and interval of convergence of the series" \sum_{n=0}^\infty \frac{x^n}{n!} Homework Equations Ratio Test The Attempt at a Solution \lim{\substack{n\rightarrow \infty}} |x/n+1| (I can't seem to get the |x/n+1| to move up where it should be)...

Find the geometric power series for the given function: f(x)=6/(2-x) c=1 I am stumped on this one. I've tried for an hour on this one with no luck. Could someone help?

I'm trying to find the sum of this: \[ \sum\limits_{n = 0}^\infty {( - 1)^n nx^n } \] This is what I have so far: \[ \begin{array}{l} \frac{1}{{1 - x}} = \sum\limits_{n = 0}^\infty {x^n } \\ \frac{1}{{(1 - x)^2 }} = \sum\limits_{n = 0}^\infty {nx^{n - 1} } =...

I need help finding the radius & interval of convergence of the following power series: The sum from n=0 to infinity of... (2-(n)^(1/2)) * (x-1)^(3n) I think the ratio test is supposed to work, but I narrow the limit of the test down to 1/|x-1|^3 < 1, and this doesn't make sense to me. Is...

I was hoping someone could check my work: Find the maclaurin power series for the function: a. f(x)=1/(4x^2+1) b. f(x)= \int e^-x^2 dx For a I got (-1)^n*2nx^n. For b I don't know where to start.

I was wondering if someone could check my work: Find the geometric power series representation of f(x)=ln(1+2x), c=0 I get \\sum_{n=0}^ \\infty2(-2x)^n+1 on -1/2<x<1/2

another power series question... I tried it for awhile but it just got out of hand and the amount of numbers got unbearable. Q. The increase in resistance of strip conductors due to eddy currents at power frequencies is given by : X = yt divided by 2 (sinh yt + sin yt divided by cosh yt -...

I need to express the sinh-1(x) as a power series in terms of powers of x. I have written the expression as x=sinhy and expanded the sinhy using the exponential series to give x = y+(1/3)y^3+(1/5)y^5+... I guess I need to expand the y=sinh-1(x) and compare or equate the coefficients. If this...

Let be the powe series: f(x)=\sum_{n=0}^{\infty}a(n)x^{n} then if f(x) is infinitely many times differentiable then for every n we have: n!a(n)=D^{n}f(0) (1) of course we don't know if the series above is of the Taylor type, but (1) works nice to get a(n) at least for finite n.

I need some help with power series. I can't remember how to find a power series center around a point. example question: y"-xy'-y=0, x=1 I don't how to start this.

Hi, Power Series' were not covered in my cal II class, so I don't know how to solve these. Is there a certain way to solve these? Find the Radiusof convergence and open disk of convergence of the power series: \frac{n^2}{2n+1}(z+6+2i)^n I don't know how to latex the summation but it is...

Im trying to determine the recurrence relation of the following differential equation : x(x-2)y'' + (1-x)y' + xy =0 about the regular singular point x =2. I've tried rewriting the DE as (x-2+2)(x-2)y'' -(x-2+1)y' +(x-2+2)y =0, but it doesn't seem to work. any ideas?

Suppose that \sum_{n=0}^{\infty} c_{n}x^{n} converges when x=-4 and diverges when x=6 . What can be said about the convergence or divergence of the following series? (a) \sum_{n=0}^{\infty} c_{n} (b) \sum_{n=0}^{\infty} c_{n}8^{n} (c) \sum_{n=0}^{\infty} c_{n}(-3)^{n} (d)...

I have a function, \frac{x^3}{e^x-1} The question than says expand the denominator as a power series in e^{-x}.I don't understand this question. How do I start doing that? It is not suggesting to approximate \frac{1}{e^x-1} as e^{-x} is it?

Sorry if this is in the wrong forum, I think it's right but I'm not totally sure. I'm (slowly) working my way through Roger Penrose's brilliant book Road to Reality. Recently I finished chapter four (Magical Complex Numbers) and one thing had me confused. How does he go about getting those...

Question: Find a power series solution in powers of x for the following differential equation xy' - 3y = k My attempt: Assume y = \sum_{m=0}^{\infty} a_m x^m So, xy' = \sum_{m=0}^{\infty}m a_m x^m xy'-3y-k=0 implies \sum_{m=0}^{\infty}m a_m x^m - 3\sum_{m=0}^{\infty} a_m x^m - k...

I just need a hint or something to see where I start. I'm at a loss for a beginning. Consider the non-homogenous equation y'' + xy' + y = x^2 +2x +1 Find the power series solution about x=0 of the equation and express your answer in the form: y=a_0 y_1 + a_1 y_2 + y_p where a_0 and...

I just need a hint or something to see where I start. I'm at a loss for a beginning. Consider the non-homogenous equation y'' + xy' + y = x^2 +2x +1 Find the power series solution about x=0 of the equation and express your answer in the form: y=a_0 y_1 + a_1 y_2 + y_p where a_0 and...

Hi All, I have this here power series which is complex valued \frac{2n+1}{2^n} i^n = \frac{4}{25} + \frac{22}{25}i My task is to prove that this power series has the above mentioned sum. To do this I separate the sum into two sums S_0 + S_1 = (i/2)^n + 2* (i/2)*n*(i/2)^{(n-1)}...

Given: y'+2xy=0 Find: Write sereis as an elementary function My solution so far: y=[Sum n=0, to infinity]C(sub-n)*x^n y'=[Sum n=1, to infinity]n*C(sub-n)*x^(n-1) y' can be transformed into: =[Sum n=0, to infinity](n+1)*C(sub-n+1)*x^n ([Sum n=0, to infinity](n+1)*C(sub-n+1)*x^n)...

I solved the differential equation for theta portion of the hydrogen wave function using a power series solution. I got a sub n+2 = a sub n ((n(n+1)-C)/(n+2)(n+1)). I then truncated the power series at n = l to get C= l(l+1). I know need to use the recursion formula I found to find the l =...

Use a four-term Taylor approximation for e^h , for h near 0 , to evaluate the following limit. lim (e^h-1-h-h^2/2)/h^3 h->0 i know that e^h = 1+h+h^2/2+h^3/3+h^4/4... therefore, I say that e^h-1-h-h^2/2 = h^3/3+h^4/4... (h^3/3+h^4/4...)/h^3 is approximately = 1/3 but its wrong...

I can't seem to get these power series to match up so that I can solve the equation...heres my work:

This is an initial value problem. y''=y'+y, y(0)=0 and y(1)=1 I am solving it by deriving the Fibonnacci power series. The sum of F_n from n=0 to infinity is 0,1,1,2,3,5,8,13 of Fibonnacci numbers defined by F_o = 0, F_1 =1. F_n = F_n-2 + F_n-1 for n>1. Here is what I have attempted so far...

Hello everyone! I remember doing these in calc II, but forgot 99% of it. Here is the question: Find the interval of convergence for the given power series. http://cwcsrv11.cwc.psu.edu/webwork2_files/tmp/equations/5e/119a4cd07ce5d3f6f673477ed169641.png The series is convergent from x = ...

may i know how to solve this ques:find the power series representation for arctan (x) i know that arctan (x) = integ 1/(1 + x^2) but then from here i don't know how to continue. pls help...

Can someone please explain some steps of a worked example. Q. Find the power series solutions about z = 0 of 4zy'' + 2y' + y = 0. (note: y = y(z)) Writing the equation in standard form: y'' + \frac{1}{{2z}}y' + \frac{1}{{4z}}y = 0 Let y = z^\sigma \sum\limits_{n = 0}^\infty...