What is Maclaurin: Definition and 198 Discussions

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. A

    Which Little "o" to use for Maclaurin expansion?

    I have calculated it and got the answer but for the first equation with the division the little o is (x^3), I believe and for the equation being multiplied by sin, little o is (x^4) For my answer do I add little o(x^4)?
  2. hilbert2

    A Analytical function from numerable point set

    Sometimes there are functions that are initially defined for only integer values of the argument, but can be extended to functions of real variable by some obvious way. An example of this is the factorial ##n!## which is extended to a gamma function by a convenient integral definition. So, if I...
  3. T

    MHB Maclaurin Limit of lnx: \frac{1}{2}

    Hello, I'm supposed to calculate the limit of this: \lim_{{x}\to{1}}\left(\frac{x}{x-1}-\frac{1}{\ln x}\right) Combining the fractions: \frac{x}{x-1}-\frac{1}{\ln x} = \frac{x\ln x-x+1}{(x-1)\ln x} The substitute u=x-1 \implies x=1+u then gives...
  4. Mr Davis 97

    Finding Maclaurin expansion and interval of convergence

    Homework Statement Find the Maclaurin series and inverval of convergence for ##f(x) = \log (\cos x)## Homework EquationsThe Attempt at a Solution I used the fact that ##\log (\cos x) = \log (1+ (\cos x - 1))##, and the standard expansions for ##\cos x## and ##\log (x+1)## to get that...
  5. Mr Davis 97

    I Exploring Maclaurin Expansions: Log(x) vs Log(x+1)

    I am looking at examples of Maclaurin expansions for different functions, such as e^x, and sinx. But there is no expansion for log(x), only log(x+1). Why is that?
  6. S

    Evaluating limit at infinity by Maclaurin series

    Homework Statement I've begun going through Boas' Math Methods in the Physical Sciences and am stuck on problem 1.15.25. The problem is to evaluate ## \lim_{x\to \infty } x^n e^{-x} ## By using the Maclaurin expansion for ##e^{x}##. Homework Equations We know the Maclaurin expansion for the...
  7. DevonZA

    Maclaurin Series Homework: Is My Solution Correct?

    Homework Statement Note - I do not know why there is a .5 after the ampere. I think it is an error and I have asked my lecturer to clarify. Homework Equations The Attempt at a Solution f(t)=sint2 f(0)=sin(0)2=0 f'(t)=2sintcost f'(0)=sin2(0)=0...
  8. T

    MHB Why is this Maclaurin series incorrect?

    I need to find the Maclaurin series for $$f(x) = x^2e^x$$ I know $$e^x = \sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ So, why can't I do $$x^2 e^x =x^2 \sum_{n = 0}^{\infty} \frac{x^n}{n!} = \sum_{n = 0}^{\infty} \frac{x^2 x^n}{n!} $$
  9. T

    MHB Finding the function of a maclaurin series

    I need to find the function for this Maclaurin series $$1 - \frac{5^3x^3}{3!} + \frac{5^5x^5}{5!} - \frac{5^7x^7}{7!} ...$$ I can derive this sigma: $$1 + \sum_{n = 2}^{\infty} \frac{(-1)^{n - 1} 5^{2n - 1} x^{2n - 1}}{(2n - 1)!}$$ But I'm not sure how to get this function from this series.
  10. T

    MHB Finding Maclaurin series

    I need to find the maclaurin series of the function $$\frac{1}{1 - 2x}$$. I know $\frac{1}{1 - x}$ is $1 + x + x^2 + x^3 ...$ but how can I use this to solve the problem? I don't think I can just plug in $2x$ can I?
  11. physiclawsrule

    MHB Question about MacLaurin series

    Aren't the Maclaurin series an expansion of a function about 0 f(x) = f(0) + (f '(0) / 1!) * x + (f ''(0) / 2!) * x^2 + (f '''(0) / 3!) * x^3 + ...
  12. T

    MHB Finding a maclaurin series for a function with 'e'

    I need to find the Maclaurin series for $$f(x) = e^{x - 2}$$ I know that the maclaurin series for $f(x) = e^x$ is $$\sum_{n = 0}^{\infty} \frac{x^n}{n!}$$ If I substitute in $x - 2$ for x, I would get $$\sum_{n = 0}^{\infty} \frac{(x - 2)^n}{n!}$$ However, this is wrong, according to the...
  13. T

    MHB Finding Maclaurin series of a natural log function

    I need to find the Maclaurin series of this function: $$f(x) = ln(1 - x^2)$$ I know that $ln(1 + x)$ equals $$\sum_{n = 1}^{\infty}\frac{(-1)^{n - 1} x^n}{n}$$ Or, $x - \frac{x^2}{2} + \frac{x^3}{3} ...$ If I swap in $-x^2$ for x, I get: $$-x^2 + \frac{x^4}{2} - \frac{x^5}{3} +...
  14. T

    MHB Maclaurin series for natural log function

    I'm examining the Maclaurin series for $f(x) = ln(x + 1)$. It is fairly straightforward but there are a few details I'm not getting. So: $$ ln(x + 1) = \int_{}^{} \frac{1}{1 + x}\,dx$$ which equals: $A + x - \frac{x^2}{2}$ etc. or $A + \sum_{n = 1}^{\infty}(-1)^{n - 1}\frac{x^n}{n}$ I'm...
  15. T

    MHB Finding Maclaurin series of a function

    I need to find the Maclaurin series for this function: $$f(x) = (1 - x)^{- \frac{1}{2}}$$ And I need to find $f^n(a)$ First, I need the first few derivatives: $$f'(x) ={- \frac{1}{2}} (1 - x)^{- \frac{3}{2}}$$ $$f''(x) ={ \frac{3}{4}} (1 - x)^{- \frac{5}{2}}$$ $$f'''(x) ={- \frac{15}{8}}...
  16. S

    Taylor polynom of f(x)=1/(√1-e3x)

    Hello, I can't find solution for Maclaurin (Taylor a=0) polynom of function: f(x)=1/(√1-e3x). Could you help me please? Thank you so much for help Andrea
  17. NihalRi

    Maclaurin series and general calculus question

    Homework Statement This question has four parts which may follow up from each other so I incuded all the parts. The real problem I'm having is with d Consider the function f ang g given by f (x)=( e^x+[e^-x])/2 & g (x) =( [e]^x]-[e^-x])/2 a) show f'(x) = g (x) and g'(x) = f (x) b) find the...
  18. nfcfox

    Macluarin series of 2^x

    Homework Statement Find the Maclaurin series for f(x) by any method. f(x)=2^x Homework Equations d/dx(b^x)= ln(b)b^x The Attempt at a Solution Ok so I basically took the derivative about 3 or so times and came out with ∑ n=0 to ∞ of ((ln(2))^n(something has to go here))/n! This much I have...
  19. P

    Question about deriving Maclaurin Series

    Homework Statement As I've been going through examples in my textbook they are becoming increasingly lengthy to compute and thus I have resorted to using software to complete the task. For example when computing the series for ##\sin{(\ln{(1+x)})}##...
  20. P

    Maclaurin Series for ##\int_{0}^{x} \cos{t^2} \cdot dt##

    Homework Statement Find the Maclaurin series of ##\int_{0}^{x} \cos{t^2} \cdot dt ## Homework Equations 3. The Attempt at a Solution [/B] I normally have some idea how to go about solving these but for this one I just can't figure out where to start. I tried doing it with ##\int_{0}^{x}...
  21. P

    Write the Maclaurin series for (1+x)^(-1/2) as a sum

    Homework Statement Write the Maclaurin series for ##\frac{1}{(1+x)^{1/2}} ## in ##\sum## form using the binomial coefficient notation. Then find a formula for the binomial coefficients in terms of n. Homework Equations 3. The Attempt at a Solution [/B]...
  22. P

    Proving Maclaurin Series for 1st Law of Blackbody Radiation

    My homework question is about the first law of blackbody radiation. I have to prove an expansion when for KT≫ℏw. After some rewriting of the formula i have (ex-1)-1 because KT≫ℏw, x is close to zero, so i think i should use the maclaurin series. According Wolfram Alpha the series expansion is...
  23. A

    Question about Maclaurin series - calculus

    Homework Statement Find the Maclaurin series of the function https://webwork.wustl.edu/webwork2_files/tmp/equations/87/63afd4b6f3566e2a90aa420dc5d1821.png [Broken] c_3 = c_4 = c_5 = c_6 = c_7 = Homework Equations The Attempt at a Solution (8x^2)[(9x) - (9x)^3/3! + (9x)^5/5! - (9x)^7/7! +...
  24. A

    Conceptual: Are all MacLaurin Series = to their Power Series?

    Homework Statement To rephrase the question, given a power series representation for a function, like ex , and its MacLaurin Series, when I expand the two there's no difference between the two, but my question is: Is this true for all functions? Or does the Radius of Convergence have to do with...
  25. K

    Maclaurin expansion and error

    my professor told me any n times differentiable function can be approximated by macularine/taylor expansion.is that true? As far as I know, if the function is approximated at point a, the approximation is valid if we pick a point near a. however, if we assume that we picked a point of...
  26. F

    MHB Help with Maclaurin series representation

    Hello! So, I'm having a bit of a problem with an exercise in my Calculus book. I'm supposed to find the Maclaurin series representation of \frac{1+x^3}{1+x^2} and then express it as a sum. Am I really supposed to differentiate this expression a bunch of times..? That will be very...
  27. I

    MHB Find First 5 Nonzero Terms of Maclaurin Series for $e^{4x} \sqrt{1+x}$

    find the first 5 nonzero terms in maclaurin series. (might be binomial) $f(x)=e^{4x} \sqrt{1+x}$my book doesn't explain it properly and my instructor didnt explain it and I am very stuck and there's going to be one similar to this on the test. help!
  28. E

    Another maclaurin vs. taylor series question

    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...
  29. A

    MHB Can you expand and find the radius of convergence for this Maclaurin series?

    Hello. I am stuck on this question. I'd appreciate if anyone could help me on how to do this. The question: Expand the following into maclaurin series and find its radius of convergence. $\frac{2-z}{(1-z)^2}$ I know that we can use geometric series as geometric series is generally...
  30. S

    Help Determining which Maclaurin Series to use for this Problem

    Homework Statement Problem is attached in this post. Homework Equations Problem is attached in this post. The Attempt at a Solution I came up with the function (1+x)^1/n and tried to derive a maclaurin series out o fit but to no avail, I can't determine what maclaurin series to...
  31. Conservation

    Difference between Taylor and MacLaurin Series-Usage

    Hi all, I understand the numerical difference between a Taylor and Maclaurin Series; Maclaurin series is just Taylor Series about x=0. However, is there any difference between their usage? I'm guessing Taylor series may be more accurate with less terms for approximating something close to...
  32. S

    Maclaurin Series Question

    Homework Statement Problem is attached in this post. Homework Equations Problem is attached in this post. The Attempt at a Solution I used the Maclaurin Series for sin (x) and got the following series: π/10 - π^3/6,000 + ... etc. I can't find a way to simplify the series...
  33. M

    Determining which of the following is a Maclaurin Series

    Homework Statement In attached image. 2. The attempt at a solution Now, after looking at the solution, the only real conclusion I can come up with is that a Maclaurin series must have x's with non-negative integer value as the exponents, correct? This is because for the the general...
  34. A

    Maclaurin Series using Substitution

    Homework Statement Use a known Maclaurin series to compute the Maclaurin series for the function: f(x) = x/(1-4(x^2))Homework Equations 1/(1-x) = ∑x^nThe Attempt at a Solution I tried removing x from the numerator for: x ∑ 1/(1-4(x^2)), which would end up through substitution as x ∑...
  35. S

    Maclaurin series homework help

    Homework Statement the maclaurin series for f(x) is given by 1/2! - x2/4! + x4/6! - x6/8! + ... + (-1)nx2n/(2n+2)! + ... a) Let g'(x) = 1-x2 * f(x) Write the Maclaurin series for g'(x), showing the first three nonzero terms and the general term. b) write g'(x) in terms of a familiar...
  36. D

    Maclaurin series of tan (e^x -1)

    Homework Statement for this, my coefficient of x^4 which is 8/4! = 1/3 .. but the ans should be 13/24... can you tell me which part contain mistake? https://i.imgur.com/05NnrdM.jpg https://i.imgur.com/28Q9o51.jpg Homework Equations The Attempt at a Solution
  37. D

    Correcting Maclaurin Series Coefficient of x^4 | Homework Help

    Homework Statement for this question, i found that my coefficient of x^4 is wrong... after applying the maclaurin series formula, i would get the coefficient of X^4 is -5/96... but the exact ans is -1/96... can anyone check which part is wrong? Homework Equations The Attempt at a...
  38. I

    MHB Maclaurin Series for e^x Example

    Hey Guys! I'm stick on this question, I know that the summation of n=0 to infinity for x^n/n! equals e^x In the question it wants me to come up with a corresponding summation for the function x^2(e^(3x^2) - 1) … I don't know how to manipulate it to get the -1. I know i can substitute x for...
  39. MarkFL

    MHB Solving Partial Fractions & Maclaurin Series Q&A

    Here is the question: I have posted a link there to this thread so the OP can view my work.
  40. J

    Integrate e^x^2, using Maclaurin rule.

    Homework Statement I am suppost to integrate e^x^2 from 0 to 1 and such, I using Maclaurins rule, I got e^x=1+x/1!+x^2/2!+...+x^n/n!+e^(öx)*x^n+1/(n+1)!, 0<ö<1. But when I put in x^2 instead of x, I end up with a legit thing except e^(öx^2)x^n+1/(n+1)! and this is giving me e^x^2 again!
  41. S

    Maclaurin series expansion

    Homework Statement Given that ##f(x)=(1+x) ln (1+x)##. (a) Find the fifth derivative of f(x), (b) Hence, show that the series expansion of f(x) is given by ##x+\frac{x^{2}}{2} -\frac{x^{3}}{6} + \frac{x^{4}}{12} - \frac{x^{5}}{20}## (c) Find, in terms of r, an expression for the rth term...
  42. C

    Maclaurin series for f(x)= ((1-x^2)/(1+x^2))

    Hi I'm studying for an upcoming exam and I have to find the Maclaurin series for f(x)= ((1-x^2)/(1+x^2)) And I got to admit i feel stuck. I know i need to find the terms f(0) +f'(0) +f''(0)/2 etc. Frist of all I can't find the first derivative f´(x) because my TI89 calculater comes up...
  43. S

    Find Maclaurin Series for g(x) with 5 Derivs of f(x)=sec(x)

    Work out the first five derivatives of the function f(x)=sec(x), and hence deduce the Maclaurin series of g(x)=sec(x)(1+tan(x)) up to and including the term of order x^4. (Hint: why have you been asked for five derivatives of f(x)?) The Maclaurin series for function g(x) is given by...
  44. polygamma

    MHB The Euler Maclaurin summation formula and the Riemann zeta function

    The Euler-Maclaurin summation formula and the Riemann zeta function The Euler-Maclaurin summation formula states that if $f(x)$ has $(2p+1)$ continuous derivatives on the interval $[m,n]$ (where $m$ and $n$ are natural numbers), then $$ \sum_{k=m}^{n-1} f(k) = \int_{m}^{n} f(x) \ dx -...
  45. C

    Maclaurin series of an elementary function question

    The Maclaurin series expansion for ##(1+z)^\alpha## is as follows: $$(1+z)^\alpha = 1 + \sum_{n=0}^\infty \binom{\alpha}{n}z^n$$ with $$|z|<1$$ What I don't understand is why is ##|z|<1##?
  46. M

    Maclaurin Series for (x2+4)-1: Terms 1-3

    Homework Statement Determine the first three terms in the Maclaurin series for: (x2+4)-1 Homework Equations f(x)=f(a)+f'(a)(x-a)+f''(a)\frac{(x-a)^{2}}{2!}+f'''(a)\frac{(x-a)^{3}}{3!} The Attempt at a Solution So I start out with getting my primes of f(x)...
  47. L

    Finding Maclaurin series for trig function

    Homework Statement Find the Maclaurin series for (tanx)2 Homework Equations f(0)+\frac{f'(0)}{1!}x+\frac{f''(0)}{2!}x^{2}+... The Attempt at a Solution I don't see how it's reasonable to do this problem without using a computer. The derivative of (tanx)2 is 2tanxsec2x, then the...
  48. J

    Maclaurin Series for Natural Log Function

    Homework Statement Use x=-1/2 in the MacLaurin series for e^x to approximate 1/sqrt(e) to four decimal places.Homework Equations The Attempt at a Solution \sum_{n=0}^\infty \frac{x^n}{n!} = 1 + x + x^2/2 + x^3/6 + ... For this particular power series, I have: \sum_{n=0}^\infty...
  49. J

    Maclaurin series for sin^2(x)

    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...