Taylor series Definition and 480 Threads

I am just trying to clarify this point which I am unsure about: If I am asked to write out (for example) a third order taylor polynomial for sin(x), does that mean I would write out 3 terms of the series OR to the x^3 term. x-x^3/3!+x^5/5! or just x-x^3/3!Also, I have a question for the...

Homework Statement If \int_{0}^{1} f(x) g(x) \ dx converges, and assuming g(x) can be expanded in a Taylor series at x=0 that converges to g(x) for |x| < 1 (and perhaps for x= -1 as well), will it always be true that \int_{0}^{1} f(x) g(x) \ dx = \int_{0}^{1} f(x) \sum_{n=0}^{\infty}...

I have just started learning about series and I don't see the benefit of shifting the series by using some "a" other than 0? My textbook doesn't really tell the benefits it just says "it is very useful"'

How is the Taylor remainder of a series (with given Taylor expansion) expressed if you want to make a calculation with known error? e.g. if I want to calculate π to, say, 12 decimal places using the previously-derived result π=4*arctan(1) and the Taylor series for arctan(x), how will I work out...

Homework Statement Sorry if this is a dumb question, but say you have 1/(1-x) This is the form of the geometric series, and is simply, sum of, from n = 0 to infiniti, X^n. I am also trying to think in terms of Binomial Series (i.e. 1 + px + p(p-1)x/2!...p(p-1)(p-2)(p-(n-1) / n!). 1/(1-x) is...

This is a very basic question . Actually in Taylor series expansion of say "sin x" we write the expansion ... (as it is,I am not writing it) But when we are asked to write the expansion of sin(x^2) we just replace 'x' by "x^2" in the expansion of sin x. Or if asked some other function such as...

Homework Statement [/B] lim x -> 0 2. Homework Equations Taylor series for sin cos e and ln () The Attempt at a Solution I tried expanding the sine to 3-degree, and everything else 2-degree. I ended up with this: Now the problem is that WolframAlpha says it should be -6/25. Now if...

Homework Statement The goal of this problem is to approximate the value of ln 2. We will use two different approaches: (a) First, we use the Taylor polynomial pn(x) of the function f(x) = lnx centered at a = 1. Write the general expression for the nth Taylor polynomial pn(x) for f(x) = lnx...

Homework Statement Let's pretend I am given a potential energy function and nothing else. I need to find the effective spring constant for oscillation about the equilibrium point using a taylor series expansion. I can't find an example or explanation anywhere on how to do this. the potential...

Homework Statement If f(x) = x^5*cos(x^6) find f40(0) and f41(0) The Attempt at a Solution So we are supposed to get the Taylor series and use that to get the value of the derivatives I just manipulated the Taylor series for cosx to get the one for this. Would the value be the coefficient?

The problem is as the title says. This is an example we went through during the lecture and therefore I have the solution. However there is a particular step in the solution which I do not understand. Using the Taylor series we will write sin(x) as: sin(x) = x - (x^3)/6 + (x^5)B(x) and...

Have a quick question about taylor series. We covered taylor series somewhat in class, but there was a complete lack of explanation and our calculus book literally covers the topic in a single page. I understand the idea of a taylor series and how its related to a power series, but what I don't...

Homework Statement Find the taylor series representation for the following function f(x) = cos(x) in powers of x-pi Homework Equations The Attempt at a Solution [/B] I don't know what they mean by "in powers of x-pi", that's the part I'm confused with. Can somebody please explain that part...

Homework Statement A standard procedure for finding an approximate mean and variance of a function of a variable is to use a Taylor Expansion for the function about the mean of the variable. Suppose the variable is y, and that its mean and standard deviation are "u" and "o". f(y) = f(u) +...

Homework Statement Hi guys, any help on this question would be hugely appreciated. The Taylor series about 0 for the function f(x)=(1/4+x)-3/2 is f(x)=8 - 48x + 240x^2 - 1120x^3 + ... used differentiation to find the Taylor series about 0 for the function g(x)=(1/4+x)-5/2 The...

find the taylor series for $f(x)=x^4-3x^2+1$ centered at $a=1$. assume that f has a power series expansion. also find the associated radius of convergence. i found the taylor series. its $-1-2(x-1)+3(x-1)^2+4(x-1)3+(x-1)^4$ but how do i find the radius of convergence?

Hey guys, Struggling with understanding this taylor vs. maclaurin series stuff. So a few questions. Let's say that we have some function f(x). 1. By saying that we want to find the power series of f(x) and nothing else, are we implicitly stating that we are looking for a maclaurin...

hi everyone , i don't understand these steps for Taylor Expansion , it has used for state space equations the equations are the approximations for sin and cos the equation for Taylor series is ( i don't understand at all ) please help me if you can

Hello everyone, I am currently reading chapter two, section 3 of Griffiths Quantum Mechanics textbook. Here is an excerpt that is giving me some difficulty: "Formally, if we expand V(x) in a Taylor series about the minimum: V(x) = V(x_0) + V'(x_0) (x-x_0) + \frac{1}{2} V''(x_0)(x-x_0)^2...

Homework Statement Find the Taylor Series of x^(1/2) at a=1 Homework Equations i have no idea how to do the representation, i believe our professor does not want us to use any binomial coefficients The Attempt at a Solution i got the expansion and here's my attempt at the...

I'm reading a derivation and it says that the following approximation can be used: I do not under stand how Taylor's theorem allows for this approximation. Can anyone explain this a little?

I want to learn combinatorics. Please send links? If possible, can you explain now?

Homework Statement Find the Taylor series for 0.5x^2[e^x-e^(-x)] around x=0. What is the coefficient of x^n? Homework Equations e^x=∑x^n/n! The Attempt at a Solution I understand how to find the Taylor series for this equation (it being ∑[x^(2n+3)/n!]; x^3+x^5+x^7/2!+...) through...

Hi could anyone give me pointer as to where to go with part iii please?

Homework Statement F(x)=7x Determine the 13th taylor coefficient of the taylor series generated by f at x=3 Homework Equations Well, it looks like I just had to take the derivative, but by the time it gets to the 13th derivative, wouldn't the answer just be zero? The Attempt at a...

1. Homework Statement [/b] Determine the Taylor series for the function below at x=0 by computing P 5 (x) f(x)=cos(7x^2) Homework Equations I used to taylor series for cosx and replaced it with 7x^2 so i used 1-49x^4/2! +2401x^8/4!... and so on. That should be correct, my attempt...

Homework Statement Consider the PM (phase modulated) signal, s(t) = Acos(wt+x(t)) where x(t) is the information bearing signal. Assume that |x(t)|< y, which is not necessarily small. Using Taylor's series expansion, derive an estimate for the bandwidth of the PM signal s(t). Homework...

I have a equation which represents a nonlinear system.I need to linearize it to obtain a linear system.I have studied various notes and asked my teachers but they are unable to explain how the solution has been obtained.I have the solution but I want to know how it has been done.Please could...

How would I find the second-degree Taylor series for $$x^2+y^2=4$$ at $$[1, -\sqrt[]{3}]$$?

Struggling with this limit value Homework Statement Calculate lim((e^x-1)/x)^(1/sin(x)) where x\rightarrow0 Homework Equations Maclaurin series. sin(x)/x -----> 1 when x->0 (possibly) The Attempt at a Solution (e^x-1)/x)^(1/sin(x) = ((x+x^2/2+x^3H(x))/x)^(1/sin(x))...

All analitic function can be express how: f(x) = \frac{1}{0!} \frac{d^0f}{dx^0}(x_0) (x - x_0)^0 + \frac{1}{1!} \frac{d^1 f}{dx^1}(x_0) (x - x_0)^1 + \frac{1}{2!} \frac{d^2f}{dx^2}(x_0) (x - x_0)^2 + \frac{1}{3!} \frac{d^3f}{dx^3}(x_0) (x - x_0)^3 + ... that is the taylor series of the function...

I attached a picture of the problem from my online HW. I know how to solve the problem through direct differentiation, but that would too long to find the derivatives for this problem, and the problem actually suggests that I find another way. So my question is, what's the best way to solve this?

Hi everyone, Homework Statement + Homework Equations + The Attempt at a Solution

Use taylor series to show that the infinite series from n=0 of $$\frac{s^n}{n!}\frac{d^n}{dq^n}(e^{-q^2})=e^{-(q+s)^2}$$

Homework Statement Compute the first four terms of the Taylor series of \frac{1}{1+e^{z}} at z_{0} = 0 and give it's radius of convergence. Homework Equations e^{z} = \sum\frac{z^{n}}{n!} = 1 + z +\frac{z^{2}}{2!} + \frac{z^{3}}{3!} + o(z^{3}) \frac{1}{1+w} =...

Homework Statement Show that if cosΦ is replaced by its third-degree Taylor polynomial in Equation 2, then Equation 1 becomes Equation 4 for third-order optics. [Hint: Use the first two terms in the binomial series for ℓ^{-1}_o and ℓ^{-1}_i. Also, use Φ ≈ sinΦ.] Homework Equations Sorry that...

Homework Statement Determine the real number to which the series \sum^{∞}_{k=1} (2-e)^k/2^k(k!) Homework Equations I know that e^x = the series of x^k / k The Attempt at a Solution I would assume to sub in 2-e for x, but then that takes away the x.

1. Homework Statement [/b] use taylor series to evaluate lim x -> 0 of \frac{ln(x)}{(x-1)}[b] Homework Equations I know that -ln (1-x) taylor polynomial and that of ln (1+x) The Attempt at a Solution Using the basics that I know I would assume I would just make ln (1+x) = ln (x)...

Homework Statement Find the 5 jet of the following function at z=0: f(z) = \frac{sinhz}{1+exp(z^3)} Homework Equations \frac{1}{1-z}=\sum_{n=0}^\infty z^n where z=-exp(z^3) The Attempt at a Solution I have tried to multiply the series for sinhz by the series for \frac{1}{1-(-exp(z^3))} but...

Homework Statement Where does the Taylor series converge? [You do not need to find the Taylor Series itself] f(x)=... I have a few of these, so I'm mainly curious about how to do this in general. The Attempt at a Solution I haven't really made an attempt yet. If I were to make an...

Homework Statement ∫((cosX)-1)/x dx Homework Equations Taylor Series The Attempt at a Solution My approach was basically to to split the integral into two more manageable parts which gave me ∫(cosX/x)dx - ∫(1/x)dx The solutions manual did it completely differently and...

Homework Statement Obtain the Taylor series ez=e Ʃ(z-1)n/n! for 0\leq(n)<\infty, (|z-1|<\infty) for the function f(z)=ez by (ii) writing ez=ez-1e. Homework Equations Taylor series: f(z) = Ʃ(1/2\pi/i ∫(f(z)/(z-z0)n+1dz)(z-z0)n The Attempt at a Solution The first part of this...

So, I have this DE which is 2nd order, w/ variable coefficients, it goes; xy''+(x-5)y'+(x^2-4)y=0 revolving around x_0=4. I know there's a singular point at 0 and I assume to use a summation y(x)=[∞,Ʃ,n=0] a_n(x-x_0)^n pardon me I don't know how to type the summation symbol, but that's...

Homework Statement Using power series, expand ln(x + 2) about a = 0 (Taylor series) Homework EquationsThe Attempt at a Solution Is this appropriate? ln(x+2) = ln((x+1)+1) x' = x+1 ln(x'+1) = \sum_{n=0}^{\inf} \frac{(-1)^n}{n+1}(x')^{n+1} or ln(x+2) = ln(\frac{x}{2}+1) x' =...

I'm at the end of a very long Poisson Processes question, involving inhomogeneous Poisson Processes. I just need to be able to expand the following expression to be able to complete the question. exp[{(sin ∏h)/∏} -h] Would anyone please be able to provide some help, with steps please!

This is rather embarrassing, because I should have known how to do this for years. Question: Compute the Taylor Series of ##\frac{q}{\sqrt{1+x}}## about x = 0. Attempt at Solution: Term-wise, I have gotten... ##f(0)+f'(0)+f''(0)+... =...

Homework Statement \lim_{x \to 0}[\frac{\sin(\tan(x))-\tan(\sin(x))}{x^7}]Homework Equations \sin(x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!} + ... \tan(x)=x+\frac{x^3}{3}+\frac{2x^5}{15}+\frac{17x^7}{215}+ ...The Attempt at a Solution I have an idea of how to do this by replacing...

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...