Power series Definition and 644 Threads

\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}

How can I go about using a power series representation of arctan(x) to approximate Pi to five digits?

Homework Statement The problem is: (x2 - 4) y′′ + 3xy′ + y = 0, y(0) = 4, y′(0) = 1 Homework Equations Existence of power series: y = \sum c(x-x0)^n or y = (x-x0)^r\sum c(x-x0)^n The Attempt at a Solution I know the point x=2 is an ordinary point of the differential...

how can i find a power series for this integral? \int cos(x^3)

Homework Statement determine the radius of convergence of the given power series \sum^{inf}_{n=1}\frac{n!x^n}{n^n} Homework Equations The Attempt at a Solution I did the ratio test then I had to take the 'ln' but, my answer is this |e|<1 for the series to converge. It...

Homework Statement Suppose that the power series \Sigma[a]_{n}[/tex]z^{n} and \Sigma b_{n} z^{n} havr radii of convergence R! and R2 respictively. Prove that the radius of convergence of the multiplication is at least R1 * R2 Homework Equations The Attempt at a Solution I...

Homework Statement let an= \sum^{k=1}_{n} 1/\sqrt{k} what is the radius of convergence of \Sigma\suma^{n=1}_{infinity} a_{n}x^n i tired including the an term into the x^n equation then i got stuck.. help please 2. Suppose that \alpha and \beta are positive real numbers with...

Homework Statement find the first three non zero terms of a power series representation of f(x)= sinh 2x Homework Equations The Attempt at a Solution seems easy enough do I just substitute 2x for x? so sinh 2x= 2x + 8x3/3! + 32x5/5!

Homework Statement using the power series method (centered at t=0) y'+t^3y = 0 find the recurrence relation Homework Equations y= \sum a_{n}t^{n} from n=0 to infinity y'= \sum na_{n}t^{n-1} from n=1 to infinity The Attempt at a Solution I went through and solved by putting the...

Homework Statement Find the radius of convergence of \sumn!*xn from n=0 to \infty Homework Equations The Attempt at a Solution I did the ratio test and was able to get it down to abs(x) * lim as n approaches \infty of abs(n+1). It seems to me that the radius of convergence...

Im given the following power series: \sum (x-3)^n/ n I determined that the radius of convergence is R=1 and the interval of convergence is [2, 4) They ask what values of x for which series converges absolutely? and values of x for which series converges conditionally? From what i...

Homework Statement The firstthing I need to note is this is for a HISTORY of math course, so we have to use non modern techniques in most cases, some not. In other words, thequestion describes how to solve them. I'm also on a compyuter with a terrible keyboard so I'm doing my best. 1)...

Power Series ArcTan? Homework Statement Let f be the function given by f(t) = 4/(1+t^2) and G be the function given by G(x)= Integral from 0 to x of f(t)dt. A) Find the first four nonzero terms and the general term for the power series expansion of f(t) about x=0. B) Find the first four...

how can i find a power series representation for a function like f(x)= ln(1+7x)?

how can i find a power series representaion of d/dx (1/x-9)

Homework Statement y'' + x2y' + xy = 0 Homework Equations using power series: y'' = \Sigmacnn(n-1)xn-2 (n = 2 -> infinity) y' = \Sigmacn(n-1)xn-1 (n = 1 -> infinity) y = \Sigmacnxn (n = 0 -> infinty) The Attempt at a Solution by setting the above equation with its...

I just had this question on my exam and I was wondering whether my method of calculating it was right: I was meant to find sum to n terms of this: 1, (4-a)x, (7-a^2)x^2, (10-a^3)x^3, (13-a^4)x^4... Would the correct way to go about it be expand it: 1, 4x-ax. 7x^2-a^2x^2... and...

Homework Statement Use power series to estimate \int_0^1 \cos(x^2)dx with an error no greater than 0.005Homework Equations Lagrange Error Formula \frac{f^{(n+1)}}{(n+1)!}(x-a)^{(n+1)} The Attempt at a Solution My original attempt was to find the series for \cos(x^2) , integrate it, and...

Homework Statement evaluate ∑ n^2.x^n where 0<x<1 Homework Equations The Attempt at a Solution let a_n = n^2 and c=0 then radius of convergence, R=1 hence the series convergences when |x|<1 let f(x) = ∑ n^2.x^n then f'(x) = ∑ n^3.x^n-1 for n=0 to infinity then f'(x) = ∑...

Homework Statement Consider the power series Σanxn = 1+2x+3x2+x3+2x4+3x5+x6+… in which the coefficients an=1,2,3,1,2,3,1,... are periodic of period p=3. Find the radius of convergence. Homework Equations The Attempt at a Solution My attempt at a solution was to first state...

(Moderator's note: thread moved from "General Math") Hi. I am confused with this question. I tried two different ways to solve it, but I got different answers for each way. The question is "Determine the coefficient of x^100 in the pwer series form of (1+x+x^2)/((1-x^3)^2)" First, I tried...

Homework Statement If A is a diagonal matrix with the diagonal entries a1, a2, ..., an, use the power series to prove that exp(At) is a diagonal matrix with the entries exp(a1t), exp(a2t), ..., exp(ant). Homework Equations The Attempt at a Solution I can prove that A is...

Homework Statement Let F be a field. Consider the ring R=F[[t]] of the formal power series in t. It is clear that R is a commutative ring with unity. the things in R are things of the form infiniteSUM{ a_n } = a_0 + a_1 t + a_2 t +... b is a unit iff the constant term a_0 =/= 0...

Two Power Series questions that need to be solved urgently Homework Statement Question 1: The function f(x) = 2x (ln(1+x)) is represented as a power series. Find coefficients c2 through c6 of the power series. Question 2: Write a partial sum for the power series which represents the function...

Homework Statement Find the radius and interval of convergence of the power series: \sum_{n=1}^{\infty} \frac{x^{2n-1}}{(n+1)\sqrt{n}} Homework Equations .. The Attempt at a Solution My soltion: the ratio test will gives |x^2|=|x|^2 it converges if |x|^2 < 1 i.e. if |x|<1 i.e...

Homework Statement 1.Is there any non-trivial power series that uniformly converges in all R? (a non-trivial power series has infinite non-zero coefficients...) 2. 2. Let f(X) be defined in [a,b]. We'll define fn(x) = [nf(x)] / n where [t]=floor value of t... Check if the series (fn(x))...

\sum_{n=0}^{\infty}(n+1)(n+2)x^n

Hi, I am trying to prove something, but I need some kind of a result on the coefficients of a power series. Suppose f(z) has a power series expansion about zero (converges). What can I say about the sum of the absolute values of the coefficients? Ideally I would like to show this sum is...

We have already shown 1+ w+ w^2 =0 If w is the complex number exp(2*Pi*i/3) , find the power series for; exp(z) +exp(w*z) + exp (z*w^2) We have already shown 1+ w+ w^2 =0

How would you go about finding the power series for sqrt(x+1) by applying the square root algorithm. I can do it using binomial expansion and other formulas but I'm not familiar with the square root algorithm involving variables.

[b]1. Find a power series for F(x)= 3/4x^3-5, where c=1 [b]2. power series = 1/a-r [b]3. What I did was take a derivative to get a similar function that was easier to solve. I used 1/x^3-1. Then I found a series for that function. Which I got \sum(x^3)^n. Then I added back from my...

Homework Statement Find a third degree polynomial approximation for the general solution to the differential equation: \frac{d^{2}y}{dt^{2}} +3\frac{dy}{dt}+2y= ln(t+1) Homework Equations Power series expansion for ln(t+1) The Attempt at a Solution The system to the...

Using the power series method to solve the differential equation y'+xy=0 when y(0)=1 Write the solution in the form of a power series and then recognize what function it represents. ************************************ My answer: \sum(-1)k*[(x2k)/(2k)*k!] Is my answer correct? Is...

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Homework Statement y''+t^2*y'-y=1-t^2Homework Equations y(0)=-2 y'(0)=1 Find the first 6 coefficients C1=-2 C2=1 C3=? (-1?) C4=? C5=? C6=? The Attempt at a Solution Okay so I tried to do this but I'm not used to having anything on the RHS of the equation. I got down to...

Homework Statement "Find the radius of convergence of the power series for the following functions, expanded about the indicated point. 1 / (z - 1), about z = i. Homework Equations 1 / (1 - z) = 1 + z + z^2 + z^3 + z^4 + ... + Ratio Test: limsup sqrt(an)^k)^1/k The...

I am having trouble getting to a solution for this differential equation 2(x^2+2x)y' - y(x+1) = x^2+2x -------- 1 for a series solution, we have to assume y = \sum a_{n}x^n ---------- 2 if we divide equation 1 by x^2 + 2x , we get (x+1)/(x^2+2x) for the y term, which is where my problem...

Hey guys. I'm new here. I've been trying to figure out how to solve this problem, and I'm still confused. (-x^2 + 4x -3)* d2y/dx2 - 2(x-2) * dy/dx + 6y = 0 y(-2) = 1 dy/dx(-2) = 0 I set y = \suman(x+2)n (start at n=0, n goes to infinity) dy/dx = \sumann(x+2)n-1 (start at n=0, n...

given the infinite power series f(x)= \sum_{n=0}^{\infty}a_ {n}x^{n} if we know ALL the a(n) is there a straight formula to get the coefficients of the b(n) \frac{1}{f(x)}= \sum_{n=0}^{\infty}b_ {n}x^{n} for example from the chain rule for 1/x and f(x) could be obtain some...

Homework Statement Say f(z) = Σ(z^n), with sum from 0 to infinity Then we can say f'(z) = Σn(z^n-1), with sum from 0 to infinity (i) = Σn(z^n-1), with sum from 1 to infinity (as the zero-th term is 0)...

Homework Statement Find tan(z) up to the z^7 term, where tan(z) = sin(z)/cos(z) Homework Equations sin(z) = z - z^3/3! + z^5/5! - z^7/7! + ... cos(z) = 1 - z^2/2! + z^4/4! - z^6/6! + ... The Attempt at a Solution Hi, Seeing as sin and cos have the same power series as for when...

Homework Statement Find the power series for the function f(z) = (1-z)^-m Hint: Differentiation gives: f'(z) = m(1-z)^m-1 = m(1-z)^-1.f(z) or: zf'(z) + mf(z) = f'(z) Use the formula for differentiation of power series to determine the coefficients of the power series for f...

How do you find the power series for inverse trig functions? Can I find the power series for arcsin by manipulating the power series for sin? Thanks!

can you check my work? "power series representation" is ok I figure it out.