Taylor series Definition and 492 Threads

Homework Statement "Determine the first two non-vanishing terms in the Taylor series of \frac{1-\cos(x)}{x^2} about x = 0 using the definition of the Taylor series (i.e. compute the derivatives of the function)." So I know how compute the Taylor series about x=0; it involves finding f(0)...

Can someone please explain how the taylor series would work if x, the given value from the function, is equal to a, the value at which you expand the function? For example, let's take 1/(1-x) as an example. The taylor series for this with a=0 is Ʃ(n from 0 to infinity) x^n. But if we let...

Homework Statement The first equation on the uploaded paper converts to the last equation.Homework Equations when i substitute ln (1-u)=-u-(1/2)(u^2) into the first equation, i can get the first term in (3rd equation). but the second term of the 3rd equation ?The Attempt at a Solution I tried...

When expanding a function (for example the determinant of the space-time metric g) as a functional of a perturbation from the flat metric ##h_{\mu \nu}##, i.e. ##g_{\mu \nu} = \eta_{\mu \nu} + h_{\mu \nu} ## i would think that the thing to do is to recognize that ##g_{\mu \nu}## and thus also...

Continuing from http://www.mathhelpboards.com/f10/taylor-series-x-%3D-1-arctan-x-5056/: The discussion in that thread gave rise to a general question to me: Does not the point of differentiation change when one makes the substitution h = x -a? I like Serena affirmed this "conjecture but...

Hey forum. Is there any way one can take advantage of the Maclaurin series of \arctan (x) to obtain the Taylor series of \arctan (x) at x = 1? I attempted to obtain the series in the suggested manner but to no avail. We have \arctan (x) = \sum_{n=0}^\infty \frac { f^{(n)}(1) }{n!} (x - 1)^n...

Does one assess $x$ at $x=0$ for the entire series? (If so, wouldn't that have the effect of "zeroing" all the co-efficients when one computes?) only raising the value of $k$ by $1$ at each iteration? and thereby raising the order of derivative at each...

Calculate f^{(18)}(0) if f(x)=x^2 \ln(1+9x) if we start with ln(1+9x) and ignore x^2 we can calculate that f'(x)= \frac{9}{1+9x} <=> f'(0)=9 f''(x)= \frac{9^2}{(1+9x)^2} <=> f'(0)=9^2 . . . f^{n}(x)= \frac{9^n}{(1+9x)^n} <=> f'(0)=9^n how does it work after? Don't we have to use product rule...

Hello MHB, I am working with Taylor series pretty new for me, I am working with a problem from my book f(x)=\sin(x^3), find f^{(15)}(0). I know that \sin(x) = 1-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}...Rest How does this work now =S? Regards, |\pi\rangle

Homework Statement Evaluate the anti derivative ∫e^x^2 dx as a Taylor Series Homework Equations \frac{f^(n)(a)}{n!}(x-a)^n The Attempt at a Solution Where do I start, I am not sure I understand the question

Homework Statement I read that the taylor series was a way to approximate the a function f(x) graphically, by addition and subtraction. So say I have \frac{1}{1-x}=1+x+x^{2}+x^{3}+...+x^{n}... suppose x=3, then the left and right side of the equation can't possibly equal the same thing...

Here is the question: Here is a link to the question: How to Find the Taylor series for the function f(x)=(1)/(x) centered at a=-3? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.

1. a. Find Taylor series generated by ex2 centered at 0. b. Express ∫ex2dx as a Taylor series. 2. For part a, I just put the value of "x2" in place of x in the general form for the e^x Taylor series: ex: 1 + x + x2/2! + x3/3! + ... ex2: 1 + x2 + x4/2! + x6/3! + ... For part b...

Homework Statement Find the Taylor series of f(x) = x2ln(1+2x2) centered at c = 0. Homework Equations Taylor Series of f(x) = ln(1+x) is Ʃ from n=1 to ∞ of (-1)n-1xn/n The Attempt at a Solution I have worked the problem to (-1)n4nx2n/n I am not sure where to go from here...

Homework Statement Find the Taylor Series for f(x) = 1/(1-6) centered at c=6 Homework Equations ∞ Ʃ Fn(a)(x-a)/n! n=0 The Attempt at a Solution I believe that the nth derivative of 1/(1-6x) is (-6)n-1n!/(1-6x)n+1 So i figured that the taylor series at c=6 would be...

Homework Statement f(x)=\frac{4x}{(4+x^{2})^{2}}Homework Equations \frac{1}{1-x} = \sum x^{n} The Attempt at a Solution How am I supposed to use that equation to solve the main problem. I have the solution but I don't understand how to do any of it. My professor is horrible, been on...

Homework Statement Let g(x) = \frac{x}{e^x - 1} = \sum_{n=0}^{\infty} \frac{B_n}{n!} x^n be the taylor series for g about 0. Show B_0 = 1 and \sum_{k=0}^{n} \binom{n+1}{k} B_k = 0 .Homework Equations The Attempt at a Solution g(x) = \sum_{n=0}^{\infty} \frac{g^{(n)}(0)}{n!} x^n , but...

Homework Statement A problem from advanced calculus by Taylor : http://gyazo.com/5d52ea79420c8998a668fab0010857cf Homework Equations ##sin(x) = \sum_{n=0}^{∞} (-1)^n \frac{x^{2n+1}}{(2n+1)!}## ##sin(3x) = \sum_{n=0}^{∞} (-1)^n \frac{3^{2n+1}x^{2n+1}}{(2n+1)!}## The Attempt at a Solution...

Homework Statement Calculate the Taylor series expansion about x=0 as far as the term in ##x^2## for the function : ##f(x) = \frac{x-sinx}{e^{-x} - 1 + ln(x+1)}## when ##x≠0## ##f(x) = 1## when ##x=0## Homework Equations Some common Taylor expansions. The Attempt at a Solution...

over a Maclaurin series? Also, how do I calculate e^0 using Maclaurin series? I'm getting 0^0.

About a week ago, I learned about linear approximation from a great youtube video, it was by Adrian Banner and the series of his lectures I think were from his book Calculus LifeSaver. I truly thought it was so beautiful and powerful a concept. Shortly I also got to know the Taylor Series and...

I am trying to establish why, I'm assuming one uses taylor series, \frac{\partial u}{\partial t}(t+k/2, x)= (u(t+k,x)-u(t,x))/k + O(k^2) I have tried every possible combination of adding/subtracting taylor series, but either I can not get it exactly or my O(k^2) term doesn't work out (it's...

Prove that if $p^T▽f(x_k)<0$, then $f(x_k+εp)<f(x_k)$ for $ε>0$ sufficiently small. I think we can expand $f(x_k+εp)$ in a Taylor series about the point $x_k$ and look at $f(x_k+εp)-f(x_k)$, but what's then? (Taylor series: $f(x_0+p)=f(x_0)+p^T▽f(x_0)+(1/2)p^T▽^2f(x_0)p+...$ => here...

I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...

Homework Statement \mu = \frac{mM}{m+M} a. Show that \mu = m b. Express \mu as m times a series in \frac{m}{M} Homework Equations \mu = \frac{mM}{m+M} The Attempt at a Solution I am having trouble seeing how to turn this into a series. How can I look at the given function...

Question: http://i.imgur.com/GsjeL.png Here is my attempt so far: http://i.imgur.com/AyOCm.png Note: I've used m where the question has used j. My attempt is based off some bad notes I took in class so the way I am trying to solve the problem may not be the best. I'm struggling to...

The Taylor Series of f(x) = exp(-x^2) at x = 0 is 1-x^2... Why is this? The formula for Taylor Series is f(x) = f(0) + (x/1!)(f'(0)) + (x^2/2!)(f''(0)) + ... and f'(x) = -2x(exp(-x^2)) therefore f'(0) = 0? Can someone please explain why it is 1-x^2?

What does it mean to calculate the Taylor series ABOUT a particular point? I understand the formula for the Taylor series but how do you solve it about a particular point for a function? It's the about the particular point that confuses me. Could someone please explain this and provide...

Why in Taylor series we have some factoriel ##!## factor. f(x)=f(0)+xf'(0)+\frac{x^2}{2!}f''(0)+... Why we have that ##\frac{1}{n!}## factor?

I'm trying to understand how to use Taylor series expansion as a method to solve complex integrals. I would appreciate someone looking over my thoughts on this. I don't know if they are right or wrong or how they could be improved. I suppose that my issue is that I don't feel confident in my...

Homework Statement compute the following limit: ## \displaystyle{\lim_{x\to +\infty} x \left((1+\frac{1}{x})^{x} - e \right)} ## The Attempt at a Solution i wanted to use the taylor expansion, but didn't know what ##x_0## would be correct, as the x goes to ## \infty##. also, i tried to...

Is there a way to find the cosine of i, the imaginary unit, by computing the following infinite sum? cos(i)=\sum_{n=0}^\infty \frac{(-1)^ni^{2n}}{(2n)!} Since the value of ##i^{2n}## alternates between -1 and 1 for every ##n\in\mathbb{N}##, it can be rewritten as ##(-1)^n##...

Homework Statement My Calc II final is tomorrow, and although we never learned it, it's on the review. So I have a few examples. Some I can figure out, some I cant. Examples: f(x)=sinh(x), f(x)=ln(x+1) with x0=0, f(x)=sin(x) with x0=0, f(x)=1/(x-1) with x0=4 The only one of those that I was...

I recently thought to myself about how a slight modification to the taylor series of e^x, which is, of course: \sum_{n=0}^\infty \frac{x^n}{n!} would change the equation. How would changing this to: \sum_{n=0}^\infty \frac{x^{n/2}}{\Gamma(n/2+1)} change the equation? Would it still be...

let nε Z. the polylogarithm functions are a family of functions, one for each n. they are defined by the following taylor series: Lin(x)= Ʃ xk/kn 1.calculate the radius of convergence [b]3. when i attempted this part, i couldn't use theratio or root test, so by comparison i got R=∞...

Hello Everyone! Suppose $f(x)$ can be written as $f(x)=P_n(x)+R_n(x)$ where the first term on the RHS is the Taylor polynomial and the second term is the remainder. If the sum $\sum _{n=0} ^{\infty} = c_n x^n$ converges for $|x|<R$, does this mean I can freely write $f(x)=\sum _{n=0} ^{\infty}...

Homework Statement One uses the approximation sin(x) = x to calculate the oscillation period of a simple gravity pendulum. Which is the maximal angle of deflection (in degree) such that this approximation is accurate to a) 10%, b) 1%, c) 0.1%. You can estimate the accuracy by using the next...

Consider the function $$f(z)=e^{\frac{1}{1-z}}$$ It has an essential singularity at $z_0=1$ and hence it can be expanded in a Laurent series at $z_0$. But I'm interested in Taylor expansion. The function is analytic in the unit open disc at the origin, so I'm looking for $a_n$ where...

Find the Taylor series for cosx and indicate why it converges to cosx for all x in R. The Taylor series for cosx can be found by differentiating sum_{k=0}^{\infty} \frac{(-1)^k (x^{2k+1})}{(2k+1)!} on both sides... But I'm not sure what the question means by "why it converges to cosx for...

Homework Statement Homework Equations We just learned basic Taylor Series expansion about C, f(x) = f(C) + f'(C)(x - C) + [f''(C)(x - C)^2]/2 + ...The Attempt at a Solution Well the previous few questions involved finding the limit of the function and the derivative of the function as X...

Hello all, Recently I've found something very interesting concerning Taylor series. It's a graphical representation of a second order error bound of the series. Here is the link: http://www.karlscalculus.org/l8_4-1.html My question is: is it possible to represent higher order error bounds...

Hello, What polynomial should I use for finding Taylor series for 28^(3/5) up to the first order? I mean, aside x^3 around 2. Any suggestions?

Homework Statement Okay, first there is an explanation of the Taylor Series equation. This I don't have a problem with. Then, we have this: Consider the power series 2 - (2/3)x + (2/9)x^2 - (2/27)x^3. What rational function does this power series represent? Homework Equations / The Attempt...

OK... "A power series can be differentiated or integrated term by term over any interval lying entirely within the interval of convergence" When i do term by term differentiaion or t-by-t integration of a series though, am i making use of this fact? Does this come into play later in a...

Show that abs[ sin (x) - 6x/(6+x^2) ] <= x^5/24, for all x in [0,2] I tried to use the sine function taylor expansion but I get stuck

Homework Statement Derive the Derive the two variable second order Taylor series approximation, below, to f(x,y) = x^3 + y^3 – 7xy centred at (a,b) = (6,‐4) f(x,y) ≈ Q(x,y) = f(a,b) + \frac{∂f}{∂x}| (x-a) + \frac{∂f}{∂x}|(y-b) + \frac{1}{2!}[\frac{∂^2f}{∂x^2}| (x-a)^2 + 2\frac{∂^2f}{∂x∂y}\...

I need to calculate \sum_{n=0}^{∞}x^{(2^n)} for 0≤x<1. It doesn't resemble any basic taylor series, so I have no idea how to sum it up. Any hint, or the resulting formula? This series comes from a physical problem, so I suppose (if I didn't make a mistake) that the series is sumable, and...

We just had a lecture on power series today (Taylor and McLaurin's) and I had a couple of questions: What does it mean for an expansion to be "around the origin"? I thought that the expansion provided an approximation to the original function at all points for which the function was defined...

Using taylor series expansion to prove gravitational potential energy equation, GMm/r=mgh at distances close to the earth. R= radius of the Earth h= height above surface of the Earth m= mass of object M= Mass of the earth U = - GmM/(R + h) = - GmM/R(1+ h/R) = - (GmM/R)(1+ h/R)^-1 do a...

Homework Statement If I take a function f(x) and its taylor series, then will the infinite series give me the value of the function at any x value or will it only give proper values for x≈a? For example, If I take a maclaurin series for a function will it give me proper values for all x...