Maclaurin Definition and 196 Threads

since the maclaurin series for sin x is alternating in sign (EQ1) so when you square it to get sin^{2}(x) (EQ2) the (-1)^{n} should become (-1)^{2n} (EQ3) which can be simplified down to (EQ4), but when i checked that series at wolframalpha the series was still alternating like: Why is that? So...

Homework Statement f(x) =ln (1-x^3) / (x^2) Homework Equations Using the maclaurin series ln (1 +x) = Ʃ (-1)^(n-1) (x^n)/(n) The Attempt at a Solution the maclaurin series for the function i get is [(-1)^(2n-1) (x)^(n)] / (n) however, the answer according to my prof is...

Calculate MacLaurin-polynom of grade 3 to function $$\cos(\ln(1+2x-3x^2))$$if i make Taylor expansion in that ln first is this correct $$\ln(1+2x-3x^2)=2x-3x^2-\frac{(2x-3x^2)^2}{2}+\frac{(2x-3x^2)^3}{3}...$$ Is that correct? Regards, $$|\pi\rangle$$

Homework Statement find the Maclaurin series and find the interval on which the expansion is valid. f(x) = ln(1-x2 ) Homework Equations The Attempt at a Solution I'm pretty confident in my skill at problems like these, except for this one I am getting an answer different from...

Edit: Never mind. Got it. Homework Statement f(x)=\frac { x }{ { (2-x) }^{ 2 } } Homework Equations The Attempt at a Solution I tried finding the first derivative, the second derivative, and so on, but it just keeps getting more complicated, so I suspect I have to use binomial series. The...

Homework Statement "Find the sum of the seires: 3 + (9/2!) + 27/3! +81/4!+ ... "Homework Equations e^x = Ʃ n=0 to inf (x^n)/n! The Attempt at a Solution =3(1 +3/2! + 9/3! + 27/4! + ... =3*Ʃ n=0 to inf( (3^n)/(n+1)!) =Ʃ n=0 to inf( (3^(n+1))/(n+1)!) . unsure what to do from here, maybe...

Hi. I have a doubt about an exercise in a book of optics. It's about Youngs double slit experiment. The exercise asks to apply the law of cosines. That part was easy, you can see in the diagram, alpha is the complementary angle for theta, it goes straight forward. What I got is this...

I have just finished a unit on constructing taylor and maclaurin polynomials and series. However I am really lost on how to find the answer to this problem that i found online for the test review and its going to be on my test, I know how to construct a maclaurin polynomial and have a vague...

Homework Statement Recall that the Maclaurin series for sin(x) is \sum\frac{(-1)^{k}x^{2k+1}}{(2k + 1)!}. Use this formula to find the Maclaurin polynomial P5(x) for f(x)=xsin(x/2). Homework Equations The Attempt at a Solution I know that to approximate sin(x/2) with the Maclaurin...

Homework Statement The question asks to estimate the remainder on the interval |x|≤ 1. f(x) is given as sinh(x). I solved the polynomial P3(x) = x + (1/6)(x3) I then went ahead and solved R3(x) up to the point shown below. R3(x) = (sinh(c)*x4)(1/24)I then don't know how to go about...

Homework Statement I'm having a bit of trouble with this Maclaurin Series question. It should be simple enough but I can't get the answer which is given as x2. It's been a while since I've done series and my being rusty is a little annoying. Hopefully someone can help :) Consider...

Homework Statement what is the maclaurin series expansion of the function (1-x)^-2 Homework Equations maclaurin series The Attempt at a Solution part of the solution is to find the n derivatives of the function to setup the series MY ANSWERS n fn(x) 0...

Homework Statement Find the maclaurin series of: f(x) = \int_{0}^{x}(e^{-t^2}-1) dt The Attempt at a Solution I know e^t = \sum_{n=0}^{∞} \frac{t^n}{n!} Simple substitution gives me: e^{-t^2} = \sum_{n=0}^{∞}\frac{(-t^2)^n}{n!} Which I rewrote as e^{-t^2} =...

Homework Statement I have the equation f(x) = \frac{\lambda^{2}}{ax^{2}}-\frac{\gamma ab}{x} What I am assigned to do is find a value of x at it's smallest, then approximate the value of the function when x - x(smallest) is much much greater than x(smallest). Homework Equations f(x) = f(0)...

Now I'm trying to get my head around this question. I just know they're going to give us a large degree question like this in the exam... Let's say: I = ∫[e^(x^2)]dx with nodes being x=0 to x=0.5 The 5th degree polynomial is 1 + x^2 + (1/2)(x^4) So my queries are: How would I go about...

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...

I have to find the Maclaurin series of: (1) f(x)=cos(x)+x, (2) g(x)= cos(x^2)+x^2 (3) h(x)=x*sin(2x). I'm stuck at the first one, I kind of understand the concept of how P(0)=f(0)+f'(0)x+(f''(0)x^2)/2+. . . What it gave me when I started calculating the value of the fn was this...

Given f(x) = xe-x2 I can differentiate once and use Leibniz to show that for n greater than 1 f(n) = -2nf(n-2) - 2xf(n-1) I want to show that the Maclaurin series for f(x) converges for all x. At x = 0, the above Leibniz formula becomes f(n) = -2nf(n-2) I know that f(0) = zero so...

1. Find the Maclaurin polynomials of order n = 0, 1, 2, 3, and 4, and then find the nth MacLaurin polynomials for the function in sigma notation. cos(∏x) 2. Here is what I did: p0x = cos (0∏) = 1 p1x = cos(0∏) - ∏sin(0∏)x = 1 p2x = cos(0∏) - ∏sin(0∏)x -\frac{∏2(cos∏x)(x2)}{2!}(...

I'm currently attempting to design a program on my ti-84 calculator (ti-nspire w/ 84 faceplate) to provide an approximation of the sin(x^2) as accurate as I would like the sum the reach. I attempted to input a formula for such, sum(seq((-1)^(Z-1)*X^(4Z-2)/(2Z-1)!, Z, 1, n, 1)), "Z" being the...

For f(x) = loge(1-x) Clarification: The question isn’t loge(1+x), it’s loge(1-x) a) Find the MacLaurin series: f(0) +f'(0)x + f''(0)/2! + f'''(0)/3!...etc My question is, to what extent do I keep applying the series to, since the series goes on forever and there are no constraints...

Homework Statement By expanding a MacLaurin Series show that E_{n}=\epsilon_{n} - \mu c^{2} = - \frac{w_{0}Z^{2}}{n^{2}}[1+\frac{\alpha^{2} Z^{2}}{n}(\frac{1}{k}-\frac{3}{4n})] Homework Equations Through a lengthy derivation I arrived at \epsilon_{n}=\frac{\mu...

Homework Statement I'm just trying to understand a few things about the Maclaurin series for e^x... So, in one case, if you have a series from 1 to infinity of [(-1)^n * 3^n ]/n!, how is it that it is equal to e^-3 - 1? I understand the e^-3 part, as -3 is simply our x value from the...

1. Find the Maclaurin series for f(x) = cos(x3) and use it to determine f6(0) 2.I know what the series expansion is. My question is, are they asking what the 6th term is with x set equal to 0? If so, all terms would be equal to zero. According to the book, the solution is -360

Homework Statement Let f(x) = \arctan(\frac{1+x}{1-x}) Find f^{2005}(0) Homework Equations I'm guessing this has to do with maclaurin's? The Attempt at a Solution ... f(x) = \pi /4 + \sum^∞_{n = 0} \frac{(-1)^n}{2n+1}x^{2n+1} \sum^∞_{n = 0}\frac{f^n(0)x^n}{n!} = \pi /4 + \sum^∞_{n = 0}...

Homework Statement It's not exactly a specific homework question, but a Putnam one. It's an integral from 0 to inf of two multiplied MacLaurin (as far as I can tell) Series, and I'm trying to figure out how to convert one of them into a recognisable function. I'm really having trouble...

Homework Statement Homework Equations The Attempt at a Solution I keep getting the answer 1- 96/4!*x^4. Can't find where I'm going wrong.

Homework Statement Find the MacLaurin series expansion of f(x)=(x^3)/(x+2). Find also the higher derivative f(10)(0) Homework Equations The Attempt at a Solution I'm not sure how to approach this question. The derivative of f(x) becomes larger and larger and I'm not sure how to...

Homework Statement the question asks to find the quadratic maclaurin polynomial for f(x) Given f(x) = x sin(x)The Attempt at a Solution i know that a maclaurin series is when a=0 in a taylor series. i did the 1st-5th derivatives of f(x) and then used the formula for taylor polynomial and set...

Homework Statement State the Maclaurin series for sinx and cosx. Hence derive the Maclaurin series for tanx. Homework Equations sin(x) = x - x3/3! + x5/5! - x7/7!... cos(x) = 1 - x2/2! + x4/4! - x6/6!... The Attempt at a Solution I know you divide the series for sinx by the...

Homework Statement Find the Maclaurin series of f(x) = x^2cos(x) Homework Equations I got the answer to be (sum from n=1 to infinity) \frac{(-1)(^n+1)x(^2n)}{(2n-2)!} and the formula for the remainder is R_n(x) = \frac{f(^n+1)(c)}{(n+1)!}x(^n+1) (I have no idea how to make those exponents...

Currently, I'm doing some self studying on series, and I'm a bit confused regarding c (the value that the series is expanded about). For example, does the Maclaurin series expansion of Sin(x) and the Taylor series of Sin(x) about c = 1 both converge to Sin(x)? If so, what does the value...

I decided to put my attempt at a solution before the question, because the "solution" is what my question is about. Homework Statement Find the rate of convergence for the following as n->infinity: lim [sin(1/n^2)] n->inf Let f(n) = sin(1/n^2) for simplicity. 2. The attempt at a...

MacLaurin Series Integration... I have to find the MacLaurin for f(x)=ln(1+x^2) So i started off by finding the derivative of the function getting \frac{2x}{1+x^2} My issue lies with the 2x in the numerator. I know how to bring the x into the series, but the two? Do I leave it on...

solved thanks

Homework Statement Assume that sin(x) equals its Maclaurin series for all x. Use the first two terms of the Maclaurin series for sin(7x^2) to approximate the integral: \int_{0}^{0.77}sin(7x^{2})\ dx The Attempt at a Solution If I understand correctly, a Maclaurin series is just a...

Homework Statement Find the first three nonzero terms in the power series representation in powers of x (ie. the maclaurin series for: (the equation in the latex image below) Homework Equations fundamental theorem of calculus, e^x = sum from n=0 to infinity of x^n/n! The Attempt...

Homework Statement Find the Maclaurin series for : [ln(1+x^2)]/x Homework Equations f(x) = \sum f^{n}(0)/n! * x^{n} f(x) = f(0) + f'(0)(x) + f''(0)(x)^2/2! + ... The Attempt at a Solution I got stuck right away, as how do I determine f(0) when you can't divide by 0...

I am studying for an exam, and I am trying to figure out: if you have something like e^(x^3), can you simply substitute x^3 into the M-series for e^x and get the M-series for e^(x^3)? Or would you have to cube the whole e^x series? I have encountered mixed responses to this question. This...

Homework Statement Suppose an object is dropped from height h above Earth where h<<R, but is large enough so g, the acceleration due to gravity, is NOT constant! Show that speed with which it hits the ground, neglecting friction, is approximately, v= sqrt2gh *(1-(h/2R)) Hint: you will need...

F(x)= (cos(2x)/(1+x^2)) Is there anyway to do this without taking a lot of derivatives and looking for a pattern?

I'm trying to calculate the following function (at x=1) with accuracy of 10^(-3). f(x)= \int^x_{0} \frac{1-cost}{t} What I've tried: f(1)=f(0)+f'(c)=1-\sum_{k=0}^\infty \frac{(-1)^nc^{2k}}{(2k)!} But now I don't know how to calculate this expression. [I know that this series is convergent...

I am having some difficulty with a homework problem I was recently assigned. The problem says to "Replace each trigonometric function with its third Maclaurin polynomial and then evaluate the function at f(0.1)" This is what I have done so far: f(x)=(x cosx- sinx)/(x-sin⁡x) 1st trig...

Homework Statement \lim_{x\to0}[\frac{1}{x^2} - \frac{\cos(x)}{\sin(x)^2}] I'm supposed to use Maclaurin series to evaluate this limit. The instructions suggest, as a hint: "First combine the fractions. Then find the first term of the denominator series and the first term of the numerator...

I have hard time to come with Maclaurin Polynomial of a given order [lets say 3] for a composite function like ln(cosx). Will appreciate help of how to approach such a problem.

Suppose that f is entire,= and that f(0)=f'(0)=f''(0)=1 (a) Write the first three terms of the Maclaurin series for f(z) (b) Suppose also that |f''(z)| is bounded. Find a formula for f(z). I believe (a) is just 1+z+(z^2)/2! however (b) I do not know where to begin.

Homework Statement http://img704.imageshack.us/f/helpppp.png/ Homework Equations The Attempt at a Solution I know e^(x) = 1 + x + x^(2)/2! + ... But if you multiply that by (x^(4))+4x^(3)) How do you know what bn and a is?

Homework Statement Find the terms through x^5 in the Maclaurin series for f(x) f(x)=tanx Homework Equations tanx=sinx/cosx Maclaurin Series for: sinx=x-x^3/3!+x^5/5!-x^7/7!... cosx=1-x^2/2!+x^4/4!-X6/6!... The Attempt at a Solution I have done tanx=sinx/cosx So I...

Let f(x)=\frac{1}{x^2+x+1} Let f(x)=\sum_{n=0}^{\infty}c_nx^n be the Maclaurin series representation for f(x). Find the value of c_{36}-c_{37}+c_{38}. After working out the fraction, I arrived at the following, f(x)=\sum_{n=0}^{\infty}x^{3n}-\sum_{n=0}^{\infty}x^{3n+1} But I dun get how to...

Homework Statement Use the MacLaurin series for e^x and ln (1+x) to show that; \frac{1}{1*2}+\frac{1}{2*3}+\frac{1}{3*4}...= 1 Homework Equations e^{x}= 1 + x + \frac{x^{2}}{2!}+\frac{x^{3}}{3!}... ln(1+x)= x - \frac{x^{2}}{2}+\frac{x^{3}}{3}... The Attempt at a Solution...