Maclaurin series Definition and 152 Threads

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...

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 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)...

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...

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...

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 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...

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?

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...

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...

Homework Statement How do I find the sum of \sum\frac{x^{2k}}{k!}? The Attempt at a Solution I tried transforming various known Taylor series, such as sin x, e^x, and so on, but they didn't fit for 2 reasons: 1. In all of them, the degree of the factor equals the power of x. i.e. if you have...

Determine the first three non-zero terms of the Maclaurin’s series expansion for: f (x) = sinh(x) Would the answer to this be: x + 1/6x^3 + 1/120x^5

This is an example given in my textbook. The final answer is 1/6. I know that sinx and 2e^x have to be replaced with their corresponding Maclaurin series. However, I'm having trouble understanding the steps they took to get the limit in a form in which it could be evaluated by substituting x=0...

I've taken maths through calc 3. I understand the Maclaurin series represents a function f(x) as a power series: \sum(c_{n}x^{n}) But how the heck did Maclaurin figure out that the series \sum(c_{n}x^{n}) could represent f(x)? I mean, that's clearly not obvious from inspection. I want to...

Let's find the Maclaurin Series of tanhx up to powers of x^5 Yeah! Good idea! I know Right, f(x) = tanh f'(x) = sech^{2}(x) f''(x) = -2sech^{2}(x)tanh(x) f''(x) = 4sech^{2}(x)tanh^{2}(x) - 2sech^{4}(x) giving f(0) = 0, f'(0) = 1, f''(0) = 0 f'''(-2) but according to my textbook...

Homework Statement Find the MacLaurin series representation for f(x)=ln|1+x^3|Homework Equations 1/(1-x) = \sumx^n = 1+x+x^2+x^3+... |x|<1The Attempt at a Solution right. so maclaurin series by default means it expands as a taylor series where x=0 f(0)= ln|1+x^3| = 0 f'(0)= 3(0)^2/(1+0^3)^1 =...

Homework Statement Find the MacLaurin series of f(x) = ((e^x) - cos (x)) / x Homework Equations e^x = (x^n)/n! cos x = ((-1)^(n) (x^(2n))) / (2n)! The Attempt at a Solution I'm just working on the numerator now, but I can't figure out how to subtract the MacLaurin series listed...

Homework Statement Assume that sin(x) equals its Maclaurin series for all x. Use the Maclaurin series for sin(3 x^2) to evaluate the integral int_0^{0.72} sin(3 x^2) dx. Your answer will be an infinite series. Use the first two terms to estimate its value. Homework Equations The Attempt at a...

Homework Statement Hi. Find the sum of the first 10 terms in the series: 1, (ln2)/1!, (ln2)^2/2!, (ln^3)/3!... Homework Equations I guess the Maclaurin series, which is e^k = 1, k/1!, (k^2)/2!, (k^3)/3!... In my case, k = ln2. The Attempt at a Solution I tried to use the sum of the first...

Homework Statement [PLAIN]http://img263.imageshack.us/img263/9336/seriesgay.jpg In the previous part of the question we had to show where the taylor expansion comes from, and calculated the maclaurin series for e^x, sin x and cos x. From that we had to prove De Moivre's theorem and so I...

Homework Statement Heres the question: write the first three nonzero terms (Maclaurin Series) \int^{1+sinx}_{1}(1/(sqrt{(1+ln(x))})dx Homework Equations The Attempt at a Solution so for the similar questions, i use maclaurin series for common functions (you can see it from...

Homework Statement Estimate sin4 accurate to five decimal places (using maclaurin series of sin) Homework Equations The Attempt at a Solution Lagrange error bound to estimate sin4° to five decimal places( maclaurin series) 4°=pi/45 radians |Rn(pi/45)<1*(pi/45)^n+1/(n+1)...

Homework Statement Maclaurin series for square root (1+x) Homework Equations The Attempt at a Solution I attempted to find the maclaurin series for the function Square root of 1+x. F(0)=1 first term= 1 F'(0)=1/2 second term= (1/2)x F''(0)=-1/4 Third term (-1/4)x^2...

Given that sinh(x) = (e^x - e^-x) / 2, find a series representation for arcsinh(x). So my book did a Maclaurin series expansion of sinh(x) = x + x^3 / 3! + x^5 / 5! + ... Then it said: the inverse will have some series expansion which we will write as arcsinh(x) = b0 + b1 x + b2 x^2 + b3...

Doing a MacLaurin Series and more! The function f is defined by f(x) = 1/(1+x^3). The MacLaurin series for f is given by 1 - x^3 + x^6 - x^9 +...+ (-1)^n(x^3n) +... which converges to f(x) for -1 < x < 1. a) Find the first three nonzero terms and the general term for the MacLaurin series...

Homework Statement Compute the 9th derivative of f(x) = \frac{\cos\left(3 x^{4} \right) - 1}{x^{7}} at x=(0) Homework Equations f(x)=\sum^{\infty}_{n=0} \frac{f^{(n)}(c)}{n!}x^n cos(x)=\sum^{\infty}_{n=0} \frac{(-1)^{n}}{(2n)!}x^{2n} The Attempt at a Solution The correct answer is...

Homework Statement Write the Machlaurin series for : f(x) = (2x)/(1+x2) Homework Equations The Attempt at a Solution I tried finding all the derivatives (aka f(x), f'(x), f''(x), etc..) but the equations started getting longer and longer and would always result in 0 when x=0...

Homework Statement Find the first four terms of the Maclaurin series for f(x) = (x+1)^-2