Series expansion Definition and 187 Threads

In mathematics, a series expansion is a method for calculating a function that cannot be expressed by just elementary operators (addition, subtraction, multiplication and division).
The resulting so-called series often can be limited to a finite number of terms, thus yielding an approximation of the function. The fewer terms of the sequence are used, the simpler this approximation will be. Often, the resulting inaccuracy (i.e., the partial sum of the omitted terms) can be described by an equation involving Big O notation (see also asymptotic expansion). The series expansion on an open interval will also be an approximation for non-analytic functions.
There are several kinds of series expansions, such as:

Taylor series: A power series based on a function’s derivatives at a single point.
Maclaurin series: A special case of a Taylor series, centred at zero.
Laurent series: An extension of the Taylor series, allowing negative exponent values.
Dirichlet series: Used in number theory.
Fourier series: Describes periodical functions as a series of sine and cosine functions. In acoustics, e.g., the fundamental tone and the overtones together form an example of a Fourier series.
Newtonian series
Legendre polynomials: Used in physics to describe an arbitrary electrical field as a superposition of a dipole field, a quadrupole field, an octupole field, etc.
Zernike polynomials: Used in optics to calculate aberrations of optical systems. Each term in the series describes a particular type of aberration.
Stirling series: Used as an approximation for factorials.

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  1. P

    Integration using series expansion

    Can't we integrate tanx/x dx using the series expansion of tan x?
  2. Mapes

    Laplace transform of a Taylor series expansion

    I'm reading a paper on tissue cell rheology ("Viscoelasticity of the human red blood cell") that models the creep compliance of the cell (in the s-domain) as J(s) = \frac{1}{As+Bs^{a+1}} where 0\leq a\leq 1. Since there's no closed-form inverse Laplace transform for this expression, they...
  3. M

    Maclaurin Series expansion of Lorentz factor

    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...
  4. C

    Solving Series Expansion of Current in Resistor/Inductor Circuit

    Homework Statement A resistor and inductor are connected in series to a battery. The current in the circuit ( in A ) is given by I = 2.7(1-e-0.1), where t is the time since the circuit was closed. By using the series for ex, find the first three terms of the expansion of this function.Homework...
  5. E

    Laplace Transform of cos(kt) using Power Series expansion

    Homework Statement The problem just states to find the Laplace Transform of cos(kt) from its power series expansion, instead of using the formula for the transform of a periodic function.Homework Equations Equation for Laplace transform of a function f(t) ->\int(e^{-st}f(t))dt Power Series...
  6. I

    Differential Equations through Power Series Expansion

    When solving diff-eq's given initial values, e.g. y'' - 2y ' + y = 0 y (0) = 0 y ' (0) = 1 Can one assume immediately that y(0) = c0 and y ' (0) = c1 ? Since these are the first 2 terms in the series? Thanks!
  7. J

    What is the Taylor Series Expansion for f(x) = (sinx)/x?

    what is the taylor's series expansion polynomial for the function f(x) = [(sinx)/x] p/s : i can't open the latex reference, sorry
  8. U

    Power series expansion of a function of x

    Homework Statement [Directions to problem] Show that the function of x gives a power series expansion on some interval centered at the origin. Find the expansion and give its interval of validity. \int_0^x e^{-t^2} dt Homework Equations The Attempt at a Solution I have...
  9. T

    Power Series expansion of hyperbolic functions

    Homework Statement power series expansion of: ((cosh x)/(sinh x)) - (1/x) Homework Equations cosh x = (1/2)(ex + e-x) sinh x = (1/2)(ex - e-x) The Attempt at a Solution what i have so far: I simplified the first part of the eq to read : e2x-1 e2x-1 now I am stuck...
  10. T

    Convergence of the surface charge density Fourier series expansion

    Homework Statement Test the convergence of the series for the surface charge density: \sum^{\infty}_{s=0}(-1)^s(4s+3)\frac{(2 s -1)!}{(2s)!} Homework Equations (2s-1)! = \frac{(2s)!}{2^s s!}; (2s)! = 2^s s! Stirling's asymptotic formula for the factorials: s! = \sqrt{2 \pi s}s^s...
  11. T

    Archived How to use series expansion to simplify

    Hi, I'm new to this forum, so if there is anything wrong in this post please forgive me, I'm not sure my post will be shown correctly, so I attached a doc file. The question is A lightly damped harmonic osillator, γ<<ω0, is driven at frequency ω. 1, Find the frequency of the...
  12. I

    Trouble with a tangent series expansion

    Okay, so I am supposed to expand ln(cos(x)) basing my calculations on tan(x) = x + (1/3)x^3 + (2/15)x^5 + (17/315)x^7 + ... and expanding out to that fourth term is sufficient. I am having a major brain meltdown, I can't seem to find any equivalencies present, and I'm sure there are. Any help...
  13. N

    Calculating Uncertainty in Mass of a Star Using Taylor Series Expansion

    Homework Statement Need to calculate fractional uncertainty f, of M (mass of a star in this case), where f is much less than one. The hint i was given was all i need to know is M \alpha d3, and use a taylor expansion to the first order in f. M = mass of a star, d = distance to star...
  14. B

    Can I Combine the Series for ln(x+1) and ln(x-1) to Expand ln((x+1)/(x-1))?

    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...
  15. W

    Fourier series (cosine series expansion?)

    Find the Fourier series of the 2Pi-periodic function f(x)= {0 , Abs(x) <= Pi/2 {Abs(x)-Pi/2, Pi/2 < Abs(x) <= Pi My attempt at a solution I have sketched the function... It equals zero between -Pi/2 and Pi/2 and it equals Pi/2 at -Pi and Pi. Then the...
  16. kreil

    Taylor series expansion of tangent

    Homework Statement find the first four nonzero terms in the power series expansion of tan(x) about a=0 Homework Equations \Sigma_{n=0}^{\infty} \frac{f^n (a)}{n!}(x-a)^n The Attempt at a Solution Well the series has a zero term at each even n (0,2,4 etc) for n=1 I got x, for...
  17. T

    Power series expansion of an exponential

    Homework Statement expand the exponential term in the equation y=2[e^{x+(x²/2)}-1] as a power series Homework Equations on wikipedia I found this... http://img297.imageshack.us/img297/1088/15139862vw6.jpg The Attempt at a Solution Do I substitute x+(x²/2) as "x" in the above...
  18. T

    Series Expansion: Proving √(y)√(1+y) - ln[√(y)+√(1+y)]=2y^(3/2)/3

    Homework Statement I want to show/prove that √(y)√(1+y) - ln[√(y)+√(1+y)] = 2y^(3/2)/3 when y<<1 by series expansion. Homework Equations √(1+y) = 1+y/2 - (y^2)/8 + ... and ln[√(y)+√(1+y)] = ln[1 + √(y) + y/2 -(y^2)/8 + ...] The Attempt at a Solution I'm thinking I sub in the...
  19. T

    Fourier series expansion of Sin(x)

    May be simple, but I'm getting problem with doing Fourier series expansion of Sin(x) for -pi\leqx\leqpi
  20. Saladsamurai

    Anyone recognize this series expansion?

    Anyone recognize this series expansion?? 1+3t+\frac{9t^2}{2!}+\frac{27t^3}{3!}+\frac{51t^4}{4!}+... I looks kind of like e^t but i am not sure how to deal with it. Can I factor something... I kind of suck at these. Someone give me a hint.
  21. P

    How to Choose Bessel Function Order for Fourier-Bessel Series Expansion?

    Hello Did an exercise and a small simulation to expand f(x)=x, defined on 0<x<3 in a Fourier-Bessel series using Bessel functions of order one that satisfy the boundary condition J_1(3\lambda)=0 and I have some questions: 1.- Is there a rule to use an specific Bessel function order to do...
  22. J

    Deriving ln(x) series

    How do you go about deriving the series expansion of ln(x)? 0 < x I got the representation at math.com but i'd still like to know how they got it. It's been a while since i did calc. iii. Thanks. John
  23. B

    Taylor Series Expansion of g(z)=1/(z^3) About z0=2

    Homework Statement z is a complex number. find the taylor series expansion for g(z)=1/(z^3) about z0= 2.in what domain does the taylor series of g converge. z0 is z subscript 0 Homework Equations The Attempt at a Solution I wrote g(z)=1/(z^3) = 1/(2+(z^3)-2) = (1/2)*1/(1+(z^3...
  24. S

    Logarithmic Series Expansion Using Taylor's/McLaurin's

    Hi there guys. My first post here. I heard this forum was really helpful so I've signed up lol.I'm trying to get to grips with using Taylor's/McLaurin's formula for series expansions...My main problem lies with expansions of Logarithmic functions.. I want to work out how to expand Logs when...
  25. L

    The Taylor series expansion for sin about z_0 = (pi/2)

    Homework Statement Expand cos z into a Taylor series about the point z_0 = (pi)/2 With the aid of the identity cos(z) = -sin(z - pi/2) Homework Equations Taylor series expansion for sin sinu = \sum^{infty}_{n=0} (-1)^n * \frac{u^{2n+1}}{(2n+1)!} and the identity as given...
  26. quasar987

    Series Expansion for V(r) in Quantum Final | Step-by-Step Solution

    Homework Statement I am ashamed to ask this, but in my quantum final, there was a little mathematically-oriented subquestion that asked to show that the function V(r)=-\frac{V_0}{1+e^{(r-R)/a}} (r in [0,infty)) can be written for r>R as V_0\sum_{n=1}^{\infty}(-1)^ne^{-n(r-R)/a}The Attempt at...
  27. M

    Finding c_n in Complex Fourier Series Expansion of $e^{i \omega t}$

    I am expanding the function f(t) = e^{i \omega t} from (-π,π) as a complex Fourier series where w is not an integer. I am stuck figuring out how the series expands with n. c_n = \frac{1}{2 \pi} \int_{-\pi}^{\pi} e^{i \omega t} e^{-int} dt Join exponentials c_n = \frac{1}{2 \pi}...
  28. E

    Can We Expand a Non-L^2 Function into an Orthonormal Basis?

    Let,s suppose we have a function f(x) which is not on L^{2} space but that we choose a basis of orthononormal functions so the coefficients: c_{n}=\int_{0}^{\infty}dxf(x)\phi_{n}(x) are finite. would be valid to expand the series into this basis in the form...
  29. B

    Maclaurin Series Expansion of 5ln(7-x)

    Represent the function 5ln(7-x) as a power series, i.e., Maclaurin series, C_0= C_1= C_2= C_3= C_4= i got C_0 = 5 ln (7-0) and i think C_1 = 5/(7-1) but its wrong the textbook says that C_1 will be the derivative of C_0 anyway... please give me some hint
  30. Q

    What function correspond to this series expansion?

    Anyone knows what function correspond to this series expansion? \begin{align} f(x)=1+x+x^2+x^3+... \end{align}
  31. N

    What did I do wrong in my Taylor Series Expansion for y=kcosh(x/k)?

    y=kcosh(x/k) = k(e^(x/w) + e^(-x/k)) y ~ k[1 + x/k + (1/2!)(x^2/k^2) + (1/3!)(x^3/k^3)] +...+ k[1 - x/k +(1/2!)(x^2/k^2) - (1/3!)(x^3/k^3) +...] All odd terms except 1 cancel out. So we are left with y = k [2 + (2/2!)(x^2/k^2) + (2/4!)(x^4/k^4) + (2/6!)(x^6/k^6) +...] I've been...
  32. M

    Taylor series expansion question

    Hi, I have a question about Taylor series: I know that for a function f(x), you can expand it about a point x=a, which is given by: f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ... but I would like to do it for f(x+a) instead of f(x), and expand it about the very same point...
  33. P

    Power Series Expansion of f(x) = \frac{10}{1+100*x^2}

    the function f(x) = \frac{10}{1+100*x^2} is represented as a power series f(x) = \sum_{n=0}^{\infty} C_nX^n Find the first few coefficients in the power series: C_0 = ____ C_1 = ____ C_2 = ____ C_3 = ____ C_4 = ____ well f(x) = \frac{10}{1+100*x^2} can be written as...
  34. T

    Nonuniqueness of power series expansion

    A couple of days ago one of my teachers mentioned (when discussing the completeness of spherical harmonics) that {1,x,x^2,x^3,...} forms an overcomplete basis for (a certain class of) functions. This implies that a power series expansion of a function is not unique. And you can for instance...
  35. Y

    How can I create a power series expansion for x^2 / (1+x^2)?

    I was wondering how to create a power series expansion for the function x^2 / (1+x^2)... I've tried using the geometric series, but somehow i got stuck. thanks.
  36. Oxymoron

    Struggling with Binomial Series Expansion? Get Help Here!

    I have been doing some questions on Binomial Series expansion and have been stuck on this particular question for a long time and desperately need some guidance. Q) Expand (1/(sqrt(1-b^2(sin^2)x)))), where b = sin(1/2(theta)) as a binomial series. Here is what I have done so far... Let...
  37. W

    Power series expansion for Laplace transform

    [SOLVED] power series expansion for Laplace transform We are to find the Taylor series about 0 of e^t, take the tranform of each term and sum if possible. So I know the expansion of e^t is 1+x/1!+x^2/2!... x^n/n! then taking the tranform, 1/s + (1/1!)(1!/s^2) +(1/2!)(2!/s3)... and so on then...
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