Maclaurin Definition and 196 Threads

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...

Homework Statement Use the first two terms of the Maclaurin series expansion of cos(x) to solve the equation cos(x)-2x2=0 . Check its accuracy with a calculator ( is in radians). Homework Equations f(x)=f(0)+x(f'(0)) first two terms The Attempt at a Solution So I have found...

I came across a problem in my homework to construct a MacLaurin polynomial of the nth degree for \sqrt{1+x}, and had some major problems. I gave up and looked up the answer on the internet, which was fairly complex: \sum \frac{(-1)^{n}(2n)!x^{n}}{(1-2n)(n!)^{2}(4^{n})} Well, I know I...

Approximate the function f(x)=\sin(x) using the corresponding Maclaurin polynomial: P_5(x), in a bound \epsilon(0,\delta). Determine a value of \delta>0, so that the rest R_5(x) verifies |R_5(x)|<0.0005 for all x\in{\epsilon(0,\delta)} Well, the first thing that puzzles me a bit is that the...

Hi. Well, I have a problem with this one. It asks me to approximate \ln(0.7) using MacLaurin polynomial of fourth degree. And estimate the error. So I used: f(x)=\ln(x+1) f'(x)=\displaystyle\frac{1}{1+x} f''(x)=\displaystyle\frac{-1}{(1+x)^2} f'''(x)=\displaystyle\frac{2}{(1+x)^3}...

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...

Homework Statement Use the Maclaurin expansion of e^x to find the value of e correct to four decimal places. (This is not the same as simply using the first four terms of the expansion.) I did the question but i had to look up how many terms to use to be accurate to four decimal places (8)...

Suppose f(x) is differentiable 2 times, can I still use Maclaurin polynomial approximations and write: f(x)=f(0)+f'(0)x+\frac {f''(0)x^2} {2!}+R_2(x) If yes why? (Lagrange and Cauchy reminder theorem are using in this case the 3'rd order derivative...)

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...

Hi, I recently sat my Maths examination and there was a Maclaurin expansion question that I made an attempt at but I think it was wrong, it would be good if I could get help on this, it's too late to be of any real help but it will help me understand where I went wrong: Obtain the first three...

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 =...

I'm trying to understand the reminder of Maclaurin polynomials http://estro.uuuq.com/0.png http://estro.uuuq.com/1.png [PLAIN]http://estro.uuuq.com/2.png [PLAIN][PLAIN]http://estro.uuuq.com/3.png [PLAIN][PLAIN]http://estro.uuuq.com/4.png Here I show few attempts to use substitution...

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 f(z) = (z + 2)/(z - 2) a) Find the Maclaurin Series for f on the doman |z| < 2. b) Find the Laurent Series for f centered at z0 = 0 on domain 2 < |Z| < inf. Homework Equations The Attempt at a Solution I'm having a hard time figuring out how (z + 2)/(z -...

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...

Random Sampe of size n from distribution with pdf f(x;p)={(lnp)^x}/px! for x=0,1,...; p>1 and 0 otherwise Find CRLB for p? My problem is finding E[x] which is somekind of maclaurin series but can't figure out which one? Please any suggestions? Thanks

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

Since ln(0) doesn't exist, this question is futile right? I am tasked with finding a Maclaurin powerseries for ln(x) and to find out how many times I have to run that series to get a accurate answer for ln(1.5). What should I do? Should I find the taylor series for ln(1.5) for should I...

Homework Statement Determine the nth order Maclaurin polynomial for 1/(1-x)2 Homework Equations The known Maclaurin polynomial series: 1/(1-x)= 1+x+x2+x3+...+xn +O(xn+1) The Attempt at a Solution I tried expanding the bottom to get 1/ (1-2x+x2)) then wrote it in the form to match...

Homework Statement Find the Maclaurin polynomial for P(6)X for 1/(1+2x2) about x=0 Homework Equations I'm pretty sure I need to use the Maclaurin polynomial for 1/(1-x)= 1+x+x2+x3+...+xn +O(xn+1)The Attempt at a Solution I simply rewrote as 1/(1-(-2x2)) then proceeded to sub and expanded...

Homework Statement f(x) = e^(x^2) * sin(x) Find the value of the 3rd derivative at x = 0. Homework Equations e^x = 1 + x + x^2/2! + ... + x^n/n! sin(x) = 1 + x^3/3! - x^5/5! + ... + x^(2n+1)/(2n+1)! * (-1)^(n-1) The Attempt at a Solution I know I should plug in the two...

Homework Statement How can I calculate it for \frac{1}{1+cos^2(x)} by using the fact that \frac{1}{1+x^2} = 1 - x^2 + x^4 - ...? Homework Equations Given in the problem. The Attempt at a Solution I tried letting u = cos(x), then \frac{1}{1+cos^2(x)} = \frac{1}{1+u^2} = 1...

Homework Statement I got a few functions I need to expand to series using Maclaurin forumlas. Homework Equations http://mathworld.wolfram.com/MaclaurinSeries.html The Attempt at a Solution So here are the ones I managed to do: f= \sqrt{1-x^2-y^2} writing it in another form...

Homework Statement Wikipedia states that the Maclaurin Series expansion of the Lorentz factor is http://en.wikipedia.org/wiki/Lorentz_factor" Homework Equations Relevant equations are all found in that article The Attempt at a Solution I don't see how this comes about. My attempt...

Homework Statement find coefficient of x^4 in the MAclaurin series for f(x)=e^sinx Homework Equations ok... so taking derivatives 4 times for this function...gave me a mess! @.@ can someone help me in simplying the derivatives...? 1. cosxe^sinx then for 2. is it...

Homework Statement The Maclaurin series for a function f is given by \sum\frac{x^n}{2n}. What is the value of f(4)(0), the fourth derivative of at x = 0? a.) 1 b.) 2 c.) 3 d.) 4 e.) 5 Homework Equations The Maclaurin Series is the infinite series centered at x = 0 with the following...

Homework Statement Find the first six nonvanishing terms in the Maclaurin series solution of the initial value problem (x^2 - 3)y''(x) + 2xy'(x) = 0 where y(0) = y0 and y'(0) = y1. Homework Equations The Attempt at a Solution Should with just something like Φ(x) such that Φ(x) =...

Homework Statement The formula for the Maclaurin Polynomial of sin(x) can be found on this page: http://www.tvalx.com/MathArticles/ExploringTaylorPolynomials/ExploringTaylorPolynomials.htm (close to the top). Find the Mauclaurin Polynomial of degree 4. Use it to estimate sin(0.5)...

Homework Statement Find the Maclaurin Series for g(x)= (4)/(4+2x+x^2) and its interval of convergence. Homework Equations I know the Maclaurin Series usually involves taking derivatives but every other problem I've done so far has had a degree that I've solved to. So, other than the general...

Homework Statement Assume that e^x equals its Maclaurin series for all x. Use the Maclaurin series for e^(-4 x^4) to evaluate the integralYour answer will be an infinite series. Use the first two terms to estimate its value. Homework Equations The Attempt at a Solution I've tried using the...

Homework Statement Find the Maclaurin series for f(x)= 1/ (1+x+X2) Homework Equations The Attempt at a Solution I think the book says I can just divide 1 by the Maclaurin series of (1+x+X2). And when i do this the original function is the answer (which makes sense). But when...

Homework Statement Compute the sixth derivative of f(x)=sin(x^2) at x=0. Hint: Maclaurin polynomial may be helpful to you. Homework Equations The taylor expansion/maclaurin expansion for sin. The Attempt at a Solution Help.

Homework Statement (1 pt) Compute the 9th derivative of: f(x) = \frac{cos(6x^4)-1}{x^7} at x=0. Homework Equations Hint: Use the MacLaurin series for f(x). The Attempt at a Solution I have tried many weird ways and cannot come up with the correct numerical answer. I've...

Find the first few terms of the Taylor Series around x=0...? of the function f(x)= {x/(e^x - 1) , x =/ 0} {1 , x=0} the function is piecewise. up to and including the term involving x^2 It says to not compute derivatives of f but to use the formula for the...

Homework Statement 5) For each of the series below, write the series in summation notation and give the first five terms of the series. Also give the radius of convergence of the series. a) Use the series for \frac{1}{1 - x} to find the Maclaurin series of f(x) = \frac{1}{(1-2x)^3}...

Hey, here's is my problem as the exam states it: A) Write out the first four non-zero terms of the Maclaurin Series for F(x) = (1+X^{7})^{-4} Give all of the coefficients in exact form, simplified as much as possible. B) Find the exact value of the 21st order derivative of F(x) =...

Homework Statement I know the sum of the Laurent series (around x=2) is equal to \frac{1}{x+3} But I can't find what the series is from this information alone. Homework Equations In the textbook, you have (for -1 < x < 1): \frac{1}{1-x} = \sum_{n=0}^{\infty}x^n and for |x|>1 I know...

Hi all, I am trying to work out a series expansion for ln ((x+1)/(x-1)). I have got the series expansion for ln(x+1) ie x- (x^2/2) + (x^3/3) - (x^4/4) ... and for ln(x-1) -x- (x^2/2) - (x^3/3) - (x^4/4) ... Can I tie these two together to get the series for ln...

I have the following problem: find the first 3 non-zero terms in the Maclaurin series for the function: e-x2 + Cos[x] I know in this case, the series behave like polynomials and I have done the following. The left expression is the first 3 terms of the e portion of the problem, and the...