Power series Definition and 629 Threads

Homework Statement http://sites.math.rutgers.edu/~ds965/temp.pdf (NUMBER 2)[/B]Homework Equations I do not understand the alternating part for the second problem and the recursive part for the first problem.The Attempt at a Solution The first answer I got was first by writing out the...

Homework Statement Homework Equations Power series ODE The Attempt at a Solution [/B] Sorry for not typing all those things out from my phone.. How can I get C1? And how can I put the solution in the required format? (I don't know how to put it in summation sign... and i cannot even solve...

Homework Statement Homework Equations Power series The Attempt at a Solution As I have to write in form of "x^2n" & "x^2n+1", I am totally have no idea with how can I go on to do the question. Those I have learned in lecture and online are mostly with only one part of summation... or two...

Homework Statement Hello, I have a general question regarding to coefficient matching when spanning some function, say , f(x) as a linear combination of some other basis functions belonging to real Hilbert space. Homework Equations - Knowledge of power series, polynomials, Legenedre...

Hi, I keep reading in multiple sources that amplifier output can be given by Vout = a0 + a1v(t) + a2v2(t) + a3v3(t) + ... + anvn(t) I've checked in three of my textbooks and there is not a clear definition (its often just stated) why this equation is used and why it works. I am not looking...

The power series $$\sum_{n = 2}^\infty \frac{(n-1)(-1)^n}{n!}$$ converges to what number? So far, I've tried using the Ratio Test and the limit as n approaches infinity equals $0$. Also since $L<1$, the power series converges by the Ratio Test.

Homework Statement If d^2/dx^2 + ln(x)y = 0[/B]Homework Equations included in attempt The Attempt at a Solution I was confused as to whether I include the power series for ln(x) in the solution. It makes comparing coefficients very nasty though. Whenever I expand for m=0 for the a0 I end...

Is it ok to assume that the entropy ##S## of an arbritary system can be written as a power series as a function of the system's internal energy ##U##? Like $$S(U) = \sum_{i=1}^{\infty}a_iU^i = a_1 U + a_2 U^2 + \ ...$$ with ##a_i \in \mathbb{R}##. What results could be obtained from such...

Hello and thank you for trying to help. In spite of the fact that this seems a very simple problem, I do not find myself able to get a solution. Here it goes: Let $$f(x)=\displaystyle \sum_{k=3}^\infty a_k \frac{x^k}{k(k-1)(k-2)}$$ and $$g(x)=\displaystyle \sum_{k=0}^\infty a_k x^k$$. Express...

According to this page: https://en.wikipedia.org/wiki/Cantor's_theorem It says: "Cantor's theorem is a fundamental result that states that, for any set A, the set of all subsets of A (the power set of A) has a strictly greater cardinality than A itself." Furthermore, it says: "Cantor's...

$\textrm{a. find the power series representation for $g$ centered at 0 by differentiation}\\$ $\textrm{ or Integrating the power series for $f$ perhaps more than once}$ \begin{align*}\displaystyle f(x)&=\frac{1}{1-3x} \\ &=\sum_{k=1}^{\infty} \end{align*} $\textsf{b. Give interval of convergence...

Hi. I have this power serie (2^n/n)*z^n that runs from n=1 to infinity, and I have to show whether it's uniform konvergence on [-1/3, 1/3] or not. I hope someone can help me with this.

Homework Statement [/B] There are three problems that I am struggling with. 1. ∑[k2(x-2)k]/[3k] 2. ∑[(x-4)n]/[(n)(-9)n] 3. ∑[2k(x-3)k]/[k(k+1)] The Attempt at a Solution On the first two I am having problems finding the end-points of the interval of convergence. I use the ratio test. 1...

Homework Statement "Find the recurrence relation in the power series solution for ##y''-xy'-y=0## centered about ##x_0=1##." Homework Equations ##y=\sum_{n=0}^\infty a_nx^n## Answer as given in book: ##(n+2)a_{n+2}-a_{n+1}-a_n=0## The Attempt at a Solution ##y=\sum_{n=0}^\infty a_n(x-1)^n##...

Homework Statement Show that ##\displaystyle \frac{1}{1+x^2} = \frac{1}{x^2} - \frac{1}{x^4} + \frac{1}{x^6} - \frac{1}{x^8} + \cdots## Homework EquationsThe Attempt at a Solution I know that the power series expansion of ##\displaystyle \frac{1}{1+x^2}## about ##x=0## is ##1-x^2 + x^4 - x^6 +...

Homework Statement Evaluate the indefinite integral as a power series. What is the radius of convergence (R)? ##\int x^2ln(1+x) \, dx## Book's answer: ##\int x^2ln(1+x) dx = C + \sum_{n=1}^\infty (-1)^n \frac {x^{n+3}} {n(n+3)}; R = 1## Homework Equations Geometric series ##\frac {1} {1-x} =...

In the power series below, I've used the ratio test and at the end I got |x-2| times infinity which is >1 so it diverges.. and in this case there is no interval of convergence because it's times inifnity.. How did he conclude that it converges at x=2??

I've 2 questions 1) Why do we take absolute of the power series? 2) I don't get why the interval of convergence is from -inifinity to +infinity. You can find the problem below.

Homework Statement \begin{equation} (1-x)y^{"}+y = 0 \end{equation} I am here but do not understand how to combine the two summations: Mod note: Fixed LaTeX in following equation. $$(1-x)\sum_{n=0}^{\infty}(n+2)(n+1)a_{n+2}x^n+\sum_{n=0}^{\infty}a_nx^n = 0$$

1. ... Expand the Eigenvalue as a power series in epsilon, up to second order: λ=[3+√(1+4 ε^2)]V0 / 2 Homework Equations I am familiar with power series, but I don't know how to expand this as one.[/B]The Attempt at a Solution :[/B] I have played around with the idea of using known power...

Homework Statement Find the power series in x-x0 for the general solution of y"-y=0; x0=3. Homework Equations None. The Attempt at a Solution Let me post my whole work:

Homework Statement Homework Equations Ratio test. The Attempt at a Solution [/B] I guess I'm now uncertain how to check my interval of convergence (whether the interval contains -2 and 2)...I've been having troubles with this in all of the problems given to me. Do I substitute -2 back...

$\tiny{206.r2.11}$ $\textsf{find the power series represntation for $\displaystyle f(x)=\frac{x^7}{3+5x^2}$ (state the interval of convergence), then find the derivative of the series}$ \begin{align} f(x)&=\frac{x^7}{3}\implies\frac{1}{1-\left(-\frac{5}{3}x^2\right)}&(1)\\...

$\textsf{a. Find the first four nozero terms of the Maciaurin series for the given function} \\$ \begin{align} f^0(x)&=\ln{ (6 x + 1)} &\therefore f^0(a)&=0\\ f^1(x)&=\frac{6}{(6 x + 1)} &\therefore f^1(a)&=6\\ f^2(x)&= \frac{-36}{(6 x + 1)^2} &\therefore f^2(a)&=-36\\ f^3(x)&= \frac{432}{(6 x...

So, ∫x/(1-x)... can I solve this as a power series ∫(x*Σ x^n) = ∫(Σ x^(n+1))= (1/(n+2)*Σ x^(n+2))? Is this correct? I know there is other ways to do it... But should this be correct on a test? This solution is more fun..

$\tiny{s4.12.9.13}$ $\textsf{find a power series reprsentation and determine the radius of covergence.}$ $$\displaystyle f_{13}(x) =\frac{1}{(1+x)^2}=\frac{1}{1+2x+x^2}$$ $\textsf{using equation 1 }$ $$\frac{1}{1-x} =1+x+x^2+x^3+ \cdots =\sum_{n=0}^{\infty}x^n \, \, \left| x \right|<1$$...

Homework Statement The problem asks to use a substitution y(x) to turn a series dependent on a real number x into a power series and then find the interval of convergence. \sum_{n=0}^\infty ( \sqrt{x^2+1})^n \frac{2^n }{3^n + n^3} Homework Equations After making a substitution, the book...

Homework Statement Find the power series in x-x0 for the general solution of y"-y=0; x0=3. Homework Equations None. The Attempt at a Solution I'll post my work by uploading it.

Homework Statement An object of mass m is initially at rest and is subject to a time-dependent force given by F = kte^(-λt), where k and λ are constants. a) Find v(t) and x(t). b) Show for small t that v = 1/2 *k/m t^2 and x = 1/6 *k/m t^3. c) Find the object’s terminal velocity. Homework...

Homework Statement The following series are not power series, but you can transform each one into a power series by a change of variable and so find out where it converges. ∑∞0 ((3n(n+1)) / (x+1)n Homework Equations a power series is a series of the form: a0 + a1x + a2x^2 ... + ... The...

So here is the problem I am trying to solve: You can combine two (or more) convergent power series on the same interval I. Using the properties of the geometric series, find the power series of the function below. Series: f(x) = 1/(1 - x) = sigma k = 0, infinity = 1+ x + x^2 + x^3 Function...

Homework Statement Find the power series in x for the general solution of (1+2x^2)y"+7xy'+2y=0. Homework Equations None. The Attempt at a Solution I'll post my whole work.

I need to prove that for $-1 < x < 1$ $$\frac{1}{(1 - x)^2} = 1 + 2x + 3x^2 + 4x^3 ...$$ So, according to the textbook, the geometric series has a radius of convergence $R = 1$ (I'm not sure how this is true). In any case we can compare it to: $$\frac{1}{1 - x} =\sum_{n = 0}^{\infty} x^n$$...

Homework Statement Find the power series in x for the general solution of (1+x^2)y"+6xy'+6y=0. Homework Equations None. The Attempt at a Solution I got up to an+2=-an(n+3)/(n+1) for n=1, 2, 3, 4, 5, 6... a3=-2a1 a4=0 a5=3a1 a6=0 a7=-4a1 a8=0 The answer in the book says y=a0sigma from m=0 to...

consider ODE : Show that the solution to this ODE is: Can someone tell what kind of ODE is it?I thought,it's on the form of Bernoulli ODE with P(x)=0.Is it possible to still solve it by using Bernoulli Methodology?I mean by substituting u=y^1-a with a=2? Thanks

hi, If you look at my attachment you can see that the book express that for the situation of x=+,-(1/L) we need further investigation. It means being converged or diverged is not precise. I would like to ask: Is there remarkable proof that if x=+,-(1/L) convergence or divergence is not...

Homework Statement for d/dx(e^x) , the series should be start from 2 rather than 1 , right ? because when k = 1 , the circled part would become 0 , the series for (e^x) is 1 + x + ... Homework EquationsThe Attempt at a Solution

Homework Statement http://imgur.com/12LbqWL Part b Homework EquationsThe Attempt at a Solution Since it says the first four terms, not nonzero, the first four terms would be 0-(1/3-0)+2/9(x-2)-1/9(x-2)^2 I'm confused when it says I need to find these for x=2... Do I just plug in x=2 now and...

I'm trying to find the answer to a question similar to this posted it earlier but in the wrong section I think and not explained well. $$ \sum_{{\rm n}=0}^\infty \left (-\sqrt x \right )^n \ \ \rm ?$$ Find the interval of convergence? I tried using the root test and got from 0 to 1 but when I...

I had a question similar to Σ0∞ (-1)^n (x)^(n/2) and attempted to solve it using the root test getting abs(√x)<1, but I've also seen some places answer it as √abs(x)<1 so am I skipping a step.

Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...

Homework Statement Represent the function (8x)/(6+x) as a power serioes f(x)=∑cnxn Find c0 c1 c2 c3 c4 Radius of convergence R= Homework EquationsThe Attempt at a Solution I've represented this function as (8x/9)∑(-x/6)n and found I-x/6I <1 so R=6 Through pure guessing I discovered c0=0 but...

Homework Statement Hi everybody! I'm a little struggling to fully understand the idea of radius of convergence of a function, can somebody help me a little? Are some examples I found in old exams at my university: Calculate the radius of convergence of the following power series: a)...

(a) Find a power series representation for the function. I'm struggling on the decomposition of the numerator. This exercise is from chapter 8, section 6 of Th Stewart Calculus book.

Homework Statement http://imgur.com/1aOFPI7 PART 2 Homework Equations Taylor series form The Attempt at a Solution My thought process is that the answer is 3 because using the geometric series equation (1st term)/(1-R) then you can get the sum. In this case R would be x+2 where x is -2 so 0...

Homework Statement Complete the proof that ln (1+x) equals its Maclaurin series for -1< x ≤ 1 in the following steps. Use the geometric series to write down the powe series representation for 1/ (1+x) , |x| < 1 This is the part (b) of the question where in part (a)I proved that ln (1+x)...

This is from an example in Thomas's Classical Edition. The task is to find a solution to ##\frac{dy}{dx}=x+y## with the initial condition ##x=0; y=1##. He uses what he calls successive approximations. $$y_1 = 1$$ $$\frac{dy_2}{dx}=y_1+x$$ $$\frac{dy_3}{dx}=y_2+x$$ ...

Homework Statement Need to show that [a,f(a,a^\dagger]=\frac{\partial f}{\partial a^\dagger} Homework Equations [a,a^\dagger]=1 The Attempt at a Solution Need to expand f(a,a^\dagger) in a formal power series. However I don´t know how to do it if the variables don´t commute.

Very basic issue here. Using: \frac{1}{1-x} = \sum_{i=0}^{\infty} x^{i} , |x|<0 Find the power series representation and interval of convergence for: f(x) = \frac{1}{(1-3x)^{2}} We have that: \frac{d}{dx}\left[\frac{1}{1-x}\right] = \frac{1}{(1-x)^{2}} = \sum_{i=0}^{\infty} ix^{i-1} ...

Homework Statement The problem wants me to find the limit below using series expansion. ##\lim_{x \to 0}(\frac{1}{x^2}\cdot \frac{\cos x}{(\sin x)^2})## Homework EquationsThe Attempt at a Solution (1) For startes I'll group the two fractions inside the limit together ##\lim_{x...