Maclaurin Definition and 199 Threads

Maclaurin or MacLaurin is a surname. Notable people with the surname include:

Colin Maclaurin (1698–1746), Scottish mathematician
Normand MacLaurin (1835–1914), Australian politician and university administrator
Henry Normand MacLaurin (1878–1915), Australian general
Ian MacLaurin, Baron MacLaurin of Knebworth
Richard Cockburn Maclaurin (1870–1920), US physicist and educator

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  1. J

    How can the Maclaurin series for sin^2(x) be simplified?

    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...
  2. J

    Find the Maclaurin series for the following function

    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...
  3. Petrus

    MHB Calc MacLaurin Polynom Grade 3 for \cos(\ln(1+2x-3x^2))

    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
  4. F

    Find the Maclaurin series for ln(1-x^2) and its interval of validity.

    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...
  5. T

    Finding the Maclaurin series representation

    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...
  6. dwdoyle8854

    Calculate MacLaurin Series for Finding the Sum of a Series | Homework Help

    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...
  7. Telemachus

    Cosine theorem and maclaurin expansion

    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...
  8. X

    How to solve where a maclaurin series intersects a graph

    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...
  9. phosgene

    Using known Maclaurin series to approximate modification of original

    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...
  10. C

    Maclaurin remainder interval estimate

    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...
  11. C

    What is the Quadratic Maclaurin Polynomial for f(x)=x*sin(x)?

    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...
  12. I

    Simple? maclaurin series (1-x)^-2

    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...
  13. M

    Maclaurin series of a function

    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} =...
  14. X

    Help with Maclaurin series of (1/x), (1/x^2), etc

    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)...
  15. C

    How Do You Calculate Error Bounds for Maclaurin Polynomial Approximations?

    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...
  16. S

    Taylor Series and Maclaurin Series Doubt

    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...
  17. B

    Finding the MacLaurin Series of a function

    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...
  18. S

    Deducing Maclaurin series converges from Leibniz formula

    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...
  19. M

    How Do You Find the Maclaurin Polynomials for cos(πx)?

    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!}(...
  20. E

    Program for Sin(x^2) MacLaurin Series

    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...
  21. P

    MacLaurin Series and Ratio Test for f(x) = loge(1-x)

    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...
  22. A

    Relativistic Bohr Atom and MacLaurin Series

    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...
  23. F

    What are the rules for finding Maclaurin series for e^x?

    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...
  24. jasonleroy

    Maclaurin Series Help (1st Post)

    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
  25. H

    Using Maclaurin series to find 2005-order derivative

    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}...
  26. X

    Finding a function from its MacLaurin series?

    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...
  27. M

    Find the MacLaurin polynomial of degree 5 for F(x).

    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.
  28. T

    MacLaurin Expansion to Find Higher Derivative

    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...
  29. J

    Error Bound on Tangent Maclaurin Series

    Salutations! Just checking if my logic is correct. Homework Statement I need to bound the error for \tan x on [0, \frac{\pi}{2}] Homework Equations R_n(x) = \displaystyle \frac{\tan^{n+1}(\zeta)}{(n+1)!}x^{n+1} The Attempt at a Solution So...I thought that the error...
  30. K

    Finding quadratic maclaurin polynomial

    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...
  31. L

    Deriving Maclaurin series for tanx

    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...
  32. S

    Remainder for Maclaurin Series

    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...
  33. P

    Confused about Taylor and Maclaurin Series

    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...
  34. K

    Rate of Convergence for sin(1/x^2) with Maclaurin is undefined?

    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...
  35. M

    What is the MacLaurin series for f(x)=ln(1+x^2)?

    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...
  36. I

    Discover the Power of Maclaurin Series: A Comprehensive Guide

    solved thanks
  37. L

    Estimating integral with Maclaurin series

    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...
  38. B

    Maclaurin series power expansion

    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...
  39. 9

    Maclaurin Series for ln(1+x^2)/x

    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...
  40. N

    How Do You Find the Maclaurin Series for e^(x^3)?

    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...
  41. D

    Solving a Problem with Non-Constant g: Expanding a Maclaurin Series

    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...
  42. P

    Maclaurin Series: cos(2x)/(1+x^2)

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

    Leibniz criterion and Maclaurin approximations

    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...
  44. F

    Using Maclaurin Polynomials to Evaluate Trigonometric Functions at f(0.1)

    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...
  45. H

    Find the limit using Maclaurin series:

    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...
  46. estro

    Calculating Maclaurin Polynomial of 3rd Order for ln(cosx)

    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.
  47. B

    How Do You Solve Part (b) for a Bounded |f''(z)| in a Maclaurin Series 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.
  48. C

    Taylor Series and Maclaurin Series Help

    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?
  49. T

    Find the Maclaurin Series for tanx

    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...
  50. K

    The Maclaurin Series of an inverse polynomial function

    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...
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