Power series Definition and 629 Threads

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...

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...

Is it possible to use power series to find the inverse of any function in Z_2[x]?

My textbook has an example on multiplication of power series. " Multiply the geometric series x^n by itself to get a power series for 1/(1-x)^2 for |x|<1 " from this we get the c_{n}=n+1 O.K. I get that the coefficients are 1 for all n but why +1. Could someone please explain this to me...

for this problem, y``-3y`+2y=0 how do you solve it using the power series method?

Hi I"m having truoble with fnding the power series of the following::frown: 1+(x^3)/3+(x^6)/18+(x^9)/162+... Can anyone give me a hand? Also, any hints in finding a power series? Thanks !

Hi, I'm looking for a general power series for a function of F of n operators. As normal, the operators do not necessarily commute. My first guess was: F(x,p) = \sum_{i=0}^\infty \sum_{j=0}^\infty a_{ij} x^i p^j + b_{ij}p^i x^j However, I don't think this is correct as it is possible to...

I really need help with this exercise. Consider the power series \sum_{n=0}^{\infty}(-1)^n\frac{z^{2n+1}}{2n+1}. for z\in\mathbb{C}. I need to answer the following questions: a) Is the series convergent for z = 1? This is easy; just plug in z = 1 and observe that the alternating...

Hi Just had a question. Assuming tan(x) is given by a power series with coefficiants (An). How can it be shown that An = 0 whenever n is even. Thanks

How do you know what value a power series is centered at? for example this power series is centered at 0: \sum_{n=0}^\infty\frac{x^n}{n!} what makes it centered at zero? this one is centered at -1: \sum_{n=0}^\infty{(-1)}^n{(x+1)}^n the only thing i can discern is that...

If I evaluate the power serie y(x) = \sum_{n=0}^{\infty}a_nx^n at x = 0, the first term is a_0(0)^0. But 0^0 is undefined, is it not? How is this paradox solved?

Hi, I'm preparing for my calculus II final and I had a question about Power Series. I'm posting the link for the answer solution to a practice exam (it's in pdf form) and I will ask questions based on that. http://www.math.sunysb.edu/~daryl/prcsol3.pdf On problem 12 it asks for the fifth...

summation from n=1 to infinity (x-2)^n/(n*3^n). My teacher got -1 <= x<= 5 as the interval of convergence because he found that x-2/3<1 Using the ratio test i get (x-2)n/(n+1)3, consistantly. This is driving me wild. :smile:

the function f(x) = \frac{10}{1+100*x^2} is represented as a power series f(x) = \sum_{n=0}^{\infty} C_nX^n Find the first few coefficients in the power series: C_0 = ____ C_1 = ____ C_2 = ____ C_3 = ____ C_4 = ____ well f(x) = \frac{10}{1+100*x^2} can be written as...

I am suppose to find the power series solutions to some diffeqs. y'=xy The method is to assume that y=\sum_{n=0}^{\infty}{c_nx^n} is a solution to the diffeq.Then since we can differenitate term by term we have \frac{d}{dx} \left[ \sum_{n=0}^{\infty}{c_nx^n}...

Two parts to this problem. On the first part I need someone to check my work, and I need help on solving the second part. (a) Find a power series representation for f(x) = ln(1+x). \frac{df}{dx} = \frac{1}{1+x} = \frac{1}{1-(-x)} = 1-x+x^2-x^3+x^4-... \int_{n} \frac{1}{1+x} = \int_{n}...