Taylor Definition and 878 Threads

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. I

    Proof that e is irrational using Taylor series

    Homework Statement Using the equality ##e = \sum_{k=0}^n \frac{1}{k!} + e^\theta \frac{1}{(n+1)!}## with ##0< \theta < 1##, show the inequality ##0 < n!e-a_n<\frac{e}{n+1}## where ##a_n## is a natural number. Use this to show that ##e## is irrational. (Hint: set ##e=p/q## and ##n=q##)...
  2. W

    Taylor polynomial/series, series, function series

    well, i have an calculus exam tomorrow and I'm 100% gona fail. I've neglected calculus so i could study for other subjects and left only 2 days to study taylor's polynomial aproximation, series and function series, the latter two are way more complicated than i expected. good thing is i can...
  3. S

    Taylor expansion of a scalar potential field

    Consider the potential ##U(\phi) = \frac{\lambda}{8}(\phi^{2}-a^{2})^{2}-\frac{\epsilon}{2a}(\phi - a)##, where ##\phi## is a scalar field and the mass dimensions of the couplings are: ##[\lambda]=0##, ##[a]=1##, and ##[\epsilon]=4##. Expanding the field ##\phi## about the point...
  4. B

    Taylor expansion with multi variables

    I was reading a book on differential equations when this(taylor expansion of multi variables) happened. Why does it not include derivatives of f in any form? The page of that book is in the file below.
  5. nomadreid

    Interval of convergence for Taylor series exp of 1/x^2

    Homework Statement The interval of convergence of the Taylor series expansion of 1/x^2, knowing that the interval of convergence of the Taylor series of 1/x centered at 1 is (0,2) Homework Equations If I is the interval of convergence of the expansion of f(x) , and one substitutes a finite...
  6. T

    Taylor Expansion to Understanding the Chain Rule

    I don't understand this as isn't according to chain rule, . So where is the in the above derivative of F(t)? Source: http://www.math.ubc.ca/~feldman/m226/taylor2d.pdf
  7. N

    Taylor Series (Derivative question)

    I was looking at the solution for problem 6 and I am confused on taking the derivatives of the function f(x)= cos^2 (x) I took the first derivative and did get the answer f^(1) (x)= 2(cos(x)) (-sin (x)), but how does that simplify to -sin (2x)? Is there some trig identity that I am not aware...
  8. NicolasPan

    Difference between Taylor Series and Taylor Polynomials?

    Hello,I've been reading my calculus book,and I can't tell the difference between a Taylor Series and a Taylor Polynomial.Is there really any difference? Thanks in advance
  9. mr.tea

    Taylor Polynomials Homework: Solving (a,b) w/ Error <1/100

    Homework Statement In the attached file. (a,b) Homework Equations \cos(x)=\sum_{k=0}^{n}\frac{(-1)^kx^{2k}}{(2k)!} Pn- Taylor expansion of order n The Attempt at a Solution I know that in this case, in order to get an error less than 1/100, I need 18 terms/order 18(according to Wolfram...
  10. I

    Finding a taylor series by substitution

    Hello, In finding a taylor series of a function using substitution, is it possible to use substitution for known taylor series of a function ,using different centers, and still get the same result. For example, if we have the function 1/(1+(x^2)/6) is it possible to use the taylor series of...
  11. I

    MHB Is this the correct approach for using Taylor series in this problem?

    Hi there! I need a bit of help on a homework problem. The problem is about a voltage (V) across a circuit with a resistor (R) and and inductor (L). The current at time "t" is: I= (V/R)(1/e^(-RT/L) And the problem asks me to use Taylor series to deduce that I is approximately equal to (Vt/L) if...
  12. S

    Taylor expansion of the square of the distance function

    Does it make a sense to define the Taylor expansion of the square of the distance function? If so, how can one compute its coefficients? I simply thought that the square of the distance function is a scalar function, so I think that one can write $$ d^2(x,x_0)=d^2(x'+(x-x'),x_0)=d^2(x',x_0) +...
  13. R

    (Thermo) Energy as Taylor expansion

    Homework Statement [/B] I've attached a screenshot of the problem, which will probably provide much better context than my retelling. I'm having problems with parts f and g. The most relevant piece of information is: "To get used to the process of Taylor expansions in two variables, first we...
  14. S

    Riemannin generalization of the Taylor expansion

    I thought about the Taylor expansion on a Riemannian manifold and guess the Taylor expansion of ##f## around point ##x=x_0## on the Riemannian manifold ##(M,g)## should be something similar to: f(x) = f(x_0) +(x^\mu - x_0^\mu) \partial_\mu f(x)|_{x=x_0} + \frac{1}{2} (x^\mu - x_0^\mu) (x^\nu -...
  15. Devin

    Taylor Exansion Series Derivation

    My derivation of Taylor expansion. Hope someone struggling with it gets use!
  16. N

    Strange invocation of Taylor series

    Hi all, I was working through a chapter on Lagrangians when I cam across this: "Using a Taylor expansion, the potential can be approximated as ## V(x+ \epsilon) \approx V(x)+\epsilon \frac{dV}{dx} ##" Now this looks nothing like any taylor expansion I've seen before. I'm used to ## f(x)...
  17. S

    Taylor series for ##\cos^2(x)##

    Homework Statement [/B] Write cos^2(x) as a Taylor seriesHomework Equations f(x) = cos^2(x) The Attempt at a Solution I am stumped. The cosine function as a Taylor series is 1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + (x^8/8!) - (x^10/10!) + … I have to express it as cos^2(x) and I am making a...
  18. almarpa

    Euler equations in rigid body: Taylor VS Kleppner - Kolenkow

    Hello all. After reading both chapters on rigid body motion both in Kleppner - Kolenkow and Taylor books, I still do not undertand the physical meaning of Euler equations. Let me explain: In Kleppner - Kolenkow, they claim (page 321 - 322) that in Euler equations, Γ1, Γ2 and Γ3 are the...
  19. J

    Very long Taylor expansion/partial fraction decomposition

    Homework Statement I want to express the following expression in its Taylor expansion about x = 0: $$ F(x) = \frac{x^{15}}{(1-x)(1-x^2)(1-x^3)(1-x^4)(1-x^5)} $$ The Attempt at a Solution First I tried to rewrite the function in partial fractions (its been quite a while since I've last...
  20. C

    Summing Taylor Series: Tips & Tricks

    Expanding the series to the n^{th} derivative isn't so hard, however I'm having trouble with the summation. Any tips for the summation? e.g. taylor series for sinx around x=0 in summation notation is \sum^\infty_{n=0} \frac{x^{4n}}{2n!} Thanks.
  21. 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...
  22. P

    Taylor series with using geometric series

    The question is: Determine the Taylor series of f(x) at x=c(≠B) using geometric series f(x)=A/(x-B)4 My attempt to the solution is: 4√f(x) = 4√A/((x-c)-B = (4√A/B) * 1/(((x-c)/B)-1) = (4√A/-B) * 1/(1-((x-c)/B)) using geometric series : 4√f(x) = (4√A/-B) Σ((x-c)/B)n f(x)= A/B4 *...
  23. nuuskur

    Taylor Series for Square Root Function

    Homework Statement Expand ##f(x) = \sqrt{2x+1}## into a Taylor series around point ##c=1##. Find the interval of convergence. Homework EquationsThe Attempt at a Solution I do know that ##f(x) = \sum\frac{1}{n!}f^{(n)}(c)(x-c)^n## assuming the function is representable as a Taylor series. How...
  24. almarpa

    Taylor Classical Mechanics example 4.9

    Hello all. I have almost finished chapter 4 on energy in Taylor's classical mechanics book. But in the last example in this chapter I got confused. Here it is: "A uniform rigid cylinder of radius R rolls without slipping down a sloping track as shown in Figure 4.23. Use energy conservation to...
  25. SU403RUNFAST

    Taylor expanding a physics formula

    < Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown > So the original problem was that a stationary hydrogen atom changed states from excited to lower state and emitted a photon, i solved for the energy of the photon hf taking into account the kinetic...
  26. M

    MHB Taylor expansion of second order

    Hey! :o I have to find the Taylor expansion of second order of the following functions with center the given point $(x_0, y_0)$. $f(x, y)=(x+y)^2, x_0=0, y_0=0$ $f(x, y)=e^{-x^2-y^2}\cos (xy), x_0=0, y_0=0$ I have done the following: The Taylor expansion of second order of $f...
  27. C

    Taylor series expansion of an exponential generates Hermite

    Homework Statement "Show that the Hermite polynomials generated in the Taylor series expansion e(2ξt - t2) = ∑(Hn(ξ)/n!)tn (starting from n=0 to ∞) are the same as generated in 7.58*." 2. Homework Equations *7.58 is an equation in the book "Introductory Quantum Mechanics" by...
  28. D

    How do I write taylor expansion as exponential function?

    How do I write taylor expansion of a function of x,y,z (not at origin) as an exponential function? Please see the attached image. I need help with the cross terms. I don't know how to include them in the exponential function?
  29. M

    Finding Taylor Series for Exponential Functions

    Hello, For the exercises in my textbook the directions state: "Use power series operations to find the Taylor series at x=0 for the functions..." But now I'm confused; when I see "power series" I think of functions that have x somewhere in them AND there is also the presence of an n. Here...
  30. C

    Does cos(sqrt(x)) have a valid Taylor series expansion at a=0?

    Find the Taylor series about a=0 for the function F(x) = \cos(\sqrt{x}). Taylor series expansion of a function f(x) about a \sum^{\infty}_0 \frac{f^{(n)}(a)}{n!}(x-a)^n Taylor series of \cos{x} about a=0 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} \ldots From these...
  31. 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...
  32. F

    Unitary translation operator and taylor expansion

    Homework Statement I have quite a straightforward question on the taylor expansion however I will try to provide as much context to the problem as possible: ##T(a)## is unitary such that ##T(-a) = T(a)^{-1} = T(a)^{\dagger}## and operates on states in the position basis as ##T(a)|x\rangle =...
  33. almarpa

    Classical Classical mechanics: Taylor or Kleppner/Kolenkow?

    Hello all. I know both books cover some different topics, but for the topics they share, which one do you think is better? I have checked the first chapters in both books, and, for the time being, I can't decide. So, if anyone of you have used these textbooks, maybe you can give me a piece of...
  34. DrPapper

    Exploring Mary Boas' Theorem III: Analytic Functions & Taylor Series

    On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's...
  35. P

    Differentials, taylor series, and function notation

    "Expanding the taylor series for ##f(x)##.." (See picture) is this a typo? Aren't we expanding ##f(x + \Delta x)##? Also, when we evaluate ##f(x)## (coefficients in the expansion), are we assuming ##\Delta x = 0## by setting ##x + \Delta x## (argument of the function) equal to ##x##? Or are we...
  36. P

    Understanding Taylor Series: Finding the General Formula | Math Explained"

    $$f(a + x) = \sum_{k=0}^∞ \frac{f^{(k)}(a) x^k}{k!}$$ Usually written as: $$f(t) = \sum_{k=0}^∞ \frac{f^{(k)}(a) (t-a)^k}{k!}$$ Where ##t = a + x## Is the taylor expansion supposed to give the same result for all ##a##? The reason this confuses me is because this seems to suggest that ##f(1 +...
  37. M

    Expand a Function with Taylor Series: Quick & Easy

    Hi guys, Is there an easy and quick way to expand a function that I know its Taylor series about 0 to a series about some other z_0?
  38. T

    Estimate number of terms needed for taylor polynomial

    Homework Statement For ln(.8) estimate the number of terms needed in a Taylor polynomial to guarantee an accuracy of 10^-10 using the Taylor inequality theorem. Homework Equations |Rn(x)|<[M(|x-a|)^n+1]/(n+1)! for |x-a|<d. The Attempt at a Solution All I've done so far is take a couple...
  39. B

    Estimates of the remainder in Taylor's theorem

    Here is the exercise question; Use the general binomial series to get ##\sqrt{1.2}## up to 2 decimal points In the solution the ##R_1## was given as ##|R_1|\leq {\frac{1}{8}} {\frac{(0.2)^2}{2}}## But it doesn't say where this came from and comparing this with the estimate of remainder given in...
  40. N

    MATLAB Optimizing Taylor Series Approximations in Matlab for Trigonometric Functions

    I have been working on writing g a script file that will: Calculate f(x)=5sin(3x) using the Taylor series with the number of terms n=2, 5, 50, without using the built-in sum function.  Plot the three approximations along with the exact function for x=[-2π 2π].  Plot the relative true error...
  41. M

    MHB Taylor Expansion: Wondering Which is Right?

    Hey! :o I want to find the taylor expansion of $f(x, y)=x^2 (3y-2x^2)-y^2 (1-y)^2$ at the point $(0, 1)$ and I got the following: $$f(x, y)=3x^2-(y-1)^2+3x^2 (y-1)-2 (y-1)^3-2x^4-(y-1)^4$$ but a friend of mine got the following result: $$f(x, y)=3x^2-(y-1)^2+3x^2 (y-1)+3x...
  42. C

    Find Help w/ Taylor Series: (y+dy)^0.5

    help with the following taylor series: (y+dy)^0.5 Thanks
  43. eifphysics

    Taylor Series for cos(x^5) | Computing f^(90)(0) | Homework Solution

    Homework Statement Let f(x)=cos(x^5). By considering the Taylor series for f around 0, compute f^(90)(0). by the way, I don't know how super/sub script works? Homework EquationsThe Attempt at a Solution I tried to substitute x^5 into x's Tyler Series form and solve for f^(90)(0), but it gave...
  44. M

    Estimating Remainders for Taylor Series of Sin(x)

    I am just trying to clarify this point which I am unsure about: If I am asked to write out (for example) a third order taylor polynomial for sin(x), does that mean I would write out 3 terms of the series OR to the x^3 term. x-x^3/3!+x^5/5! or just x-x^3/3!Also, I have a question for the...
  45. polygamma

    Convergence of Taylor Series and Definite Integrals: Exploring the Relationship

    Homework Statement If \int_{0}^{1} f(x) g(x) \ dx converges, and assuming g(x) can be expanded in a Taylor series at x=0 that converges to g(x) for |x| < 1 (and perhaps for x= -1 as well), will it always be true that \int_{0}^{1} f(x) g(x) \ dx = \int_{0}^{1} f(x) \sum_{n=0}^{\infty}...
  46. G

    Thomas precession Goldstein/Eisberg versus Taylor/Wheeler

    I've looked at Taylor and Wheeler's Spacetime Physics Example 103 on the Thomas Precession and also the discussion of Thomas precession in Eisberg and Goldstein (3rd edition). Both treat the rotation angle gotten by the addition of 2 non-collinear velocities. The answers they get are...
  47. M

    Why Use Taylor Series Centered at x=a in Solving Differential Equations?

    I have just started learning about series and I don't see the benefit of shifting the series by using some "a" other than 0? My textbook doesn't really tell the benefits it just says "it is very useful"'
  48. A

    Calculating Taylor Series Remainder: Finding an Upper Bound for n

    How is the Taylor remainder of a series (with given Taylor expansion) expressed if you want to make a calculation with known error? e.g. if I want to calculate π to, say, 12 decimal places using the previously-derived result π=4*arctan(1) and the Taylor series for arctan(x), how will I work out...
  49. L

    Binomial vs Geometric form for Taylor Series

    Homework Statement Sorry if this is a dumb question, but say you have 1/(1-x) This is the form of the geometric series, and is simply, sum of, from n = 0 to infiniti, X^n. I am also trying to think in terms of Binomial Series (i.e. 1 + px + p(p-1)x/2!...p(p-1)(p-2)(p-(n-1) / n!). 1/(1-x) is...
  50. B

    Facing problem in analysing Taylor series expansion

    This is a very basic question . Actually in Taylor series expansion of say "sin x" we write the expansion ... (as it is,I am not writing it) But when we are asked to write the expansion of sin(x^2) we just replace 'x' by "x^2" in the expansion of sin x. Or if asked some other function such as...
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