Power series Definition and 644 Threads

Hi hi, So I worked on this problem and I know I probably made a mistake somewhere towards the end so I was hoping one of you would catch it for me. Thank you! Pasteboard — Uploaded Image Pasteboard — Uploaded Image

Hi - my sometimes surprising set-book asks to show by series expansion, that $ \frac{1}{2}ln\frac{x+1}{x-1} =coth^{-1} (x) $ I get LHS = $ x+\frac{{x}^{3}}{3}+\frac{{x}^{5}}{5}+... $, which I think $= tanh^{-1} $ but I have found different expansions for the hyperbolic inverses, so I'd...

Homework Statement Approximate the integral to 3 decimal place accuracy via power series. ##\int_0^{1/2} x^2 e^{-x^2}\, dx ## Homework EquationsThe Attempt at a Solution ##x^2 e^{-x^2} = x^2 \sum_{n=0}^\infty \frac {(-x)^{2n}}{n!} = \sum_{n=0}^\infty \frac {x^{2n+2}}{n!}## ⇒ ##\int_0^{1/2}...

Homework Statement By considering the power series (good for |x| < 1) ##\frac{1}{1-x} = \sum_{n=0}^\infty x^n = 1 + x + x^2 + x^3 + x^4 +...## Describe how to manipulate this series in some way to obtain the result: ##\sum_{n=1}^\infty nx^n = \frac{x}{(1-x)^2}## Homework Equations Maclaurin...

Hello, Im currently in a Calc II class with unfortunately a bad professor (score of 2 on RateMyProfessor), so I often have to resort to outside sources to learn. Our class is currently on Sequences and Series which has been fine up until we hit the topic of relating Power Series and Functions...

I can not find a solid explanation on this anywhere, so forgive me if this has been addressed already. Given something like y''+y'-(x^2)y=1 or y''+2xy'-y=x, how do I approach solving a differential with a power series solution when the differential does not equal zero? Would I solve the left...

Homework Statement Let ##\sum^{\infty}_{n=0} a_n(z-a)^n## be a real or complex power series and set ##\alpha = \limsup\limits_{n\rightarrow\infty} |a_n|^{\frac{1}{n}}##. If ##\alpha = \infty## then the convergence radius ##R=0##, else ##R## is given by ##R = \frac{1}{\alpha}##, where...

For this function f(x)=x^2/(1-5x). The interval of convergence is (-1/5) < x < (1/5). I tried to differentiate, but got it wrong. Could someone please help?

I know there are a number of ways to do this problem, to increment the series etc. but, would someone please be able to explain how they get the answers for this problem simply and easily ? Thanks! A screen shot is attached

So i am given (1+x)/(1-x)^2 and I have to put it into a power series. I know that 1/(1-x)= 1+x+x^2+x^3+...=∑x^n from 0 to infinity. I am having problems factoring series. I differentiate 1/(1-x). I get, 1/(1-x)^2= 1+2x+3x^2+...= ∑nX^(n-1) the sum from 1 to infinity. rewriting this equation...

Homework Statement Σ(n=0 to ∞) ((20)(-1)^n(x^(3n))/8^(n+1) Homework Equations Ratio test for Power Series: ρ=lim(n->∞) a_(n+1)/a_n The Attempt at a Solution I tried the ratio test for Power Series and it went like this: ρ=lim(n->∞) (|x|^(3n+1)*8^(n+1))/(|x|^(3n)*8^(n+2)) =20|x|/8 lim(n->∞)...

Homework Statement Find a power series representation for the function below & determine the radius of convergence. f (x)=(1+x)/(1-x)2 2.Relevant equation Shown in attached image below which is the solution the problem. 3.The attempt at a solution I'm trying to fathom the solution here...

Homework Statement It is stated in my textbook that the sum ## \sum_{0}^{\infty} 8^{-n}(x^2-1)^n ## is not a power series but can be turned into one using he substitution ##y=x^2-1## which then becomes the power series ##\sum_{0}^{\infty} 8^{-n}y^n ## They aren't offering any explanation as to...

Homework Statement Consider ##\sum\limits_{n=0}^{\infty} \frac{n+1}{(2n)!}(x+1)^{2n+1}##. Find the interval of convergence and sum of the power series. Homework EquationsThe Attempt at a Solution According to the textbook: given the power series ##\sum a_n(x-c)^n## the radius of convergence...

Homework Statement Find an expression for e^A with the powerseries shown in the image linked Homework Equations I know you have to use eigen values and eigen vectors and a diagonal matrix The Attempt at a Solution What I did was just try to actually multiply out the infinite series given. I...

Homework Statement (1+x2) y'' + 2xy' = 0 in powers of x Homework Equations y'' = \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} y' = \sum_{n=1}^{\infty} na_nx^{n-1} The Attempt at a Solution (1+x2) y'' + 2xy' = (1+x^2) \sum_{n=2}^{\infty} (n-1)na_nx^{n-2} + 2x \sum_{n=1}^{\infty} na_nx^{n-1}...

Homework Statement To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...

please help!

Homework Statement substitute the given power series below into ODE y'' -4y=0 to verify it is a solution Homework Equations y=∑ 2n xn / n! n=0 y''-4y=0 The Attempt at a Solution I have absolutely no idea how start.

Hello. I've been solving power series problems where the question asks to find 2 power series solutions. I can solve it almost all, I can find the recurrence solution... however while checking the solutions, I see some answer solutions they used c0=1 and c1=0 to find the 2 solutions, and some...

I need to prove that the complex power series $\sum\limits_{n=0}^{\infty}a_nz^n$ converges uniformly on the compact disc $|z| \leq r|z_0|,$ assuming that the series converges for some $z_0 \neq 0.$ *I know that the series converges absolutely for every $z,$ such that $|z|<|z_0|.$ Since...

Homework Statement (x+1)y'' - (x-1)y' - y = 0 centred around x=1 y(1) = 2, y'(1) = 3 The Attempt at a Solution I know I am supposed to get two power series, one with a0 and one with a1 but when I am trying to figure out a pattern, I keep getting both a0 and a1 in all of my terms. So I end up...

Hello! (Wave) The differential equation $y''+xy=0$ is given. Find the general solution of the differential equation (with the power series method). That's what I have tried: We are looking for a solution of the form $y(x)=\sum_{n=0}^{\infty} a_n x^n$, where the radius of convergence is...

I've been reading Wald's book on General Relativity and in chapter 3 he introduces and uses the so-called Hadamard's Lemma: For any smooth (i.e. C^{\infty}) function F: \mathbb{R}^{n}\rightarrow\mathbb{R} and any a=(a^{1},\ldots,a^{n})\in\mathbb{R}^{n} there exist C^{\infty} functions H_{\mu}...

Homework Statement Give an example of a power series with [itex]R=1[\itex] that converges uniformly for [itex]|z|\le 1[\itex], but such that its derived series converges nowhere for [itex]|z=1|[\itex]. Homework Equations R is the radius of convergence and the derived series is the term by term...

I need help understanding the following solution for the given problem. The problem is as follows: Given a field $F$, the set of all formal power series $p(t)=a_0+a_1 t+a_2 t^2 + \ldots$ with $a_i \in F$ forms a ring $F[[t]]$. Determine the ideals of the ring. The solution: Let $I$ be an...

Homework Statement Hi I'm stuck with the following question: Use de Moivre's Theorem and your knowledge of power series to show: 1/1(1/2^1)cos(θ)+1/2(1/2^2)cos(2θ)+1/3(1/2^3)cos(3θ)+ ... = log(2)-1/2*log(5-4cos(θ))Homework EquationsThe Attempt at a Solution I have already established the...

Homework Statement "Use power series to evaluate the function at the given point" ## ln (x+ \sqrt{1+{x^2}}) - sin x ## at ## x = 0.001 ## Homework Equations Relevant power series: A: ## ln (1+x) = \Big( \sum_{n=0}^\infty\frac{({(-1)^{n+1}}{x^n})}{n} \Big) ## B: ## {(1+x)^p} =...

1. Write ∫e^(-t^2)dt with 0<=t<=x , as power series around 0. For what values of x this series converge ? attempt at a solution: f' = e^(-x^2) => f'(0) = 1 f''= -2x*e^(-x^2) => f''(0)= 0 f'''= -2e^(−x2) +4*x^2*e^(−x^2) => f'''(0)=-2 I tried to find a general rule for the derivatives but with...

Homework Statement Consider the power series centered at a= 0: Σkx^(k+1) From 1 to infinity (a) Find its radius of convergence, R, and its interval of convergence. = DONE (b) For x in the interval (-R,R) find the sum of the power series. Help? Homework Equations N/a The Attempt at a...

Homework Statement Find two power series solutions of the differential equation about the ordinary point x = 0. Homework Equations y'' + x^2y' +xy = 0 The Attempt at a Solution Check attachment. I found my y1 and y2, the boxed in answers are the ones the book says are the answers. Can...

Homework Statement I have this function f(x) = \frac{6}{1+49x^2}, and i suppose to represent this function as a power series \displaystyle f(x) = \sum_{n=0}^\infty c_n x^n. Then i need to find the first few coefficients in the power series. Homework EquationsThe Attempt at a Solution After...

Compute $\displaystyle\lim_{n\to +\infty}\dfrac{1^p+2^p+3^p+\cdots +n^p}{n^{p+1}}.$

I'm thinking about the following function, which is a ratio of two finite power series. I'm trying to prove the monotonicity of this function, for arbitrary K. \frac{\sum_{j=0}^k [(at)^j/j!]}{\sum_{j=0}^k [(bt)^j/j!]}, and a>b>0, t>0 I know that if k goes to infinity, the function becomes an...

Homework Statement Find the power series solution of the differential equation y''-\frac{2}{(1-x)^2}y=0 around the point ##x=0##. Homework Equations y=\sum_{n=0}^\infty{}c_nx^n y'=\sum_{n=0}^\infty{}c_{n+1}(n+1)x^n y''=\sum_{n=0}^\infty{}c_{n+2}(n+2)(n+1)x^n The Attempt...

1. What if absolute convergence test gives the result of 'inconclusive' for a given power series? We need to use other tests to check convergence/divergence of the powerr series but the matter is even if comparison or integral test confirms the convergence of the power series, we don't know...

Hi. I have another question about power series. I am having problem with the summarizing of the sum (writing in $\sum_{}^{}$ form). Here is the question: Let α be a real number that is not 0. Let $$f(z)=e^{{\alpha}Ln(z+1)}$$ For integer n>0, find $$f^n(0).$$ My partial solution...

Hi. I have another question about power series. I am having problem with the summarizing of the sum (writing in $\sum_{}^{}$ form). Here is the question: Let $\alpha$ be a real number that is not 0. Let $f(z)=e^{{\alpha}Ln(z+1)}$ For integer n>0, find $f^n(0)$. My partial solution...

Hello. I need someone to explain to me how to find the centre and radius of convergence of power series. I got the working and the answers but there are some things I don't understand. $$\sum_{n=0}^{\infty}\frac{(4i)^n(z-i)^n}{(n+1)(n+2)}$$ Using the ratio test, we got...

Hello. I need someone to explain to me how to find the centre and radius of convergence of power series. I got the working and the answers but there are some things I don't understand. $$\sum_{n=0}^{\infty}\frac{(4i)^n(z-i)^n}{(n+1)(n+2)}$$ Using the ratio test, we got $$\lim_{{n}\to{\infty}}...

As it can be read here, http://en.wikipedia.org/wiki/Laplace_transform#Relation_to_power_series the Laplace transform is a continuous analog of a power series in which the discrete parameter n is replaced by the continuous parameter t, and x is replaced by exp(-s). Therefore, computing a...

I had a quick question about an expansion. Wolfram and maple have not been very useful in verifying the series. Could I do these: Centered about a=1: ##f(x) = e^{-3x^2} = e^{-3}e^{-3(x^2-1)} = \sum_{n=0}^{∞} \frac{e^{-3}(-3)^n}{n!} (x^2-1)^n## Centered about a=2: ##f(x) =...

Hello! :cool: I am looking at the exponential power series: $$\sum_{n=0}^{\infty} \frac{x^n}{n!}$$ It is $R=\displaystyle{\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{|a_n|}}}=\frac{1}{\lim_{n \to \infty} \sup \sqrt[n]{n!}}=+\infty$ So,the power series converges at $(-\infty,+\infty)$,so...

Homework Statement a) Determine power series ##\sum _{n=0}^{\infty }a_nt^n## if you know that its laplace transformation is ##-s^{-1}e^{-s^{-1}}## b) Determine function ##g## that this power series will be equal to ##J_0(g(t))##Homework Equations The Attempt at a Solution Hmmm, I am having...

Homework Statement Solve ##y^{''}+zy=0## where ##y(0)=0## and ##y^{'}(0)=1##Homework Equations ##y(z)=z^r\sum _{k=0}^{\infty } C_kz^k## The Attempt at a Solution Well firstly: ##r(r-1)+p_0r+q_0=0## where obviously ##p_0=q_0=0## so ##r_1=0## and ##r_2=1##. In general ##y(z)=\sum...

Homework Statement find the sum of the following series: \sum_{n=1}^\infty nx^{n-1} , |x|<1 Homework Equations \frac{a}{1-r} The Attempt at a Solution i know that a function representation for that series is -\frac{1}{(1-x)^2} but how is it possible to find the sum of a series with a...

Homework Statement Problem: Find the first eight coefficients (i.e. a_0, a_1, a_2, ..., a_7) of the power series expansion y = ##Σ_{n = 0}^{∞}## [##a_n## ##x^n##] of the solution to the differential equation y'' + xy' + y = 0 subject to the initial-value conditions y(0) = 0, y'(0)...

Homework Statement Find interval of convergence and radius of convergence of the following infinite series. Series from n=1 to infinity ((-3)^n * x^n) / (n*(n)^1/2)Homework Equations Ratio testThe Attempt at a Solution I've started with the ratio test and end up getting 3xn^(3/2) /...

I was asked to find sums equal to 9/25 by using the power series of y=\frac{1}{1+x^2}. First thing I did was to find the power series representation of the function: \sum_{n=0}^{\infty }(-x^2)^n Next I figured out the interval of convergence: \left \| -x^2 \right \|< 1 This meant that x...

Homework Statement hi, for the expansion of power series (logarithmitic series) , ln(1+X) , why the condition for x is between -1 and 1 which x can be 1 but x can't be -1 ? Homework Equations The Attempt at a Solution