What is Maclaurin series: Definition and 155 Discussions

In mathematics, the Taylor series of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor's series are named after Brook Taylor, who introduced them in 1715.
If zero is the point where the derivatives are considered, a Taylor series is also called a Maclaurin series, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the 18th century.
The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).

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  1. silverfury

    Is There a Trick to Simplify Taylor Series Expansion?

    I tried diffrentiating upto certain higher orders but didn’t find any way.. is there a trick or a transformation involved to make this task less hectic? Pls help
  2. EEristavi

    Taylor/Maclaurin series of a function

    Homework Statement Obtain Maclaurin Series for: f(x) = sin(x2)/x Homework Equations f(x) = ∑f(n)(c) (x-c)n / n! (for Maclaurin c = 0) The Attempt at a Solution I know that sin(x2) = x2 - (x2*3/3! +... from the final answer I see, that this is just multiplied to 1/x. This bothers me...
  3. T

    Is this question incomplete? Regarding entire functions....

    Homework Statement Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##. Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...
  4. C

    Finding a the value of 30th derivative given power series.

    Homework Statement The problem is attached as pic Homework Equations ∑(ƒ^(n)(a)(x-a)^n)n! (This is the taylor series formula about point x = 3)The Attempt at a Solution So I realized that we should be looking at either the 30th,31st term of the series to determine the coefficient. After we...
  5. 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...
  6. T

    I How Does the -1 Arise in This Series to Function Conversion?

    I ran across an infinite sum when looking over a proof, and the sum gets replaced by a function, however I'm not quite sure how. $$\sum_{n=1}^\infty \frac{MK^{n-1}|t-t_0|^n}{n!} = \frac{M}{K}(e^{K(t-t_0)}-1)$$ I get most of the function, I just can't see where the ##-1## comes from. Could...
  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 How do I find the MacLaurin series for $\frac{1}{1 - 2x}$?

    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. Jezza

    How Can the Limit of (ln(1+x))^x as x Approaches 0 Be Evaluated Correctly?

    Homework Statement \lim\limits_{x \to 0} \left(\ln(1+x)\right)^x Homework Equations Maclaurin series: \ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} + ... + (-1)^{r+1} \frac{x^r}{r} + ... The Attempt at a Solution We're considering vanishingly small x, so just taking the first term in the...
  17. faradayscat

    Differential equation with power series

    Homework Statement Solve y''+(cosx)y=0 with power series (centered at 0) Homework Equations y(x) = Σ anxn The Attempt at a Solution I would just like for someone to check my work: I first computed (cosx)y like this: (cosx)y = (1-x2/2!+x4/4!+ ...)*(a0+a1x+a2x2 +...)...
  18. 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...
  19. J

    Proof Taylor series of (1-x)^(-1/2) converges to function

    Hello, I want to prove that the taylor expansion of f(x)={\frac{1}{\sqrt{1-x}}} converges to ƒ for -1<x<1. If I didn't make a mistake the maclaurin series should look like this: Tf(x;0)=1+\sum_{n=1}^\infty{\frac{(2n)!}{(2^n n!)^2}}x^n My attempt is to use the lagrange error bound, which is...
  20. 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)})}##...
  21. 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}...
  22. 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]...
  23. 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...
  24. 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 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! + ...] I got...
  25. 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...
  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. 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...
  29. 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...
  30. S

    Finding the Solution to a Maclaurin Series for Sin(x)

    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...
  31. 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...
  32. 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 ∑...
  33. 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...
  34. 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
  35. 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...
  36. 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...
  37. 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.
  38. S

    Finding the Maclaurin Series Expansion of (1+x)ln(1+x)

    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...
  39. 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...
  40. 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...
  41. 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##?
  42. 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)...
  43. 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...
  44. 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...
  45. 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...
  46. 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...
  47. Fernando Revilla

    MHB Nick's question at Yahoo Answers (Maclaurin series)

    Here is the question: Here is a link to the question: Find the Maclaurin series? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
  48. 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...
  49. 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...