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. polygamma

    MHB Series Expansion: Show Sin^Cos x = x + O(x^3)

    Show that for small positive $x$, $$\left( \sin x \right)^{\cos x} = x -\left( 3 \log x + 1\right) \frac{x^{3}}{3!} + \Big( 15 \log^{2} x + 15 \log x + 11 \Big) \frac{x^{5}}{5!} + \mathcal{O}(x^{7})$$
  2. Hepth

    Series Expansion of 1/Polynomial

    Is there a simple way to series expand a function of the form $$ \frac{1}{\sum_{n=0}^{\infty} a_n x^n} $$ about the point ##x=0##, such that it can be expressed as another sum ##\sum_n c_n x^n##? I tried doing it by taylor expansion but I end up with a sum of sums of products of sums :)...
  3. 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...
  4. S

    What are the constants in the binomial series expansion for (1+mx)^-n?

    Homework Statement For ##n>0##, the expansion of ##(1+mx)^{-n}## in ascending powers of ##x## is ##1+8x+48x^{2}+...## (a) Find the constants ##m## and ##n## (b) Show that the coefficient of ##x^{400}## is in the form of ##a(4)^{k}##, where ##a## and ##k## are real constants. Homework...
  5. dexterdev

    Which of the signals is not the result of fourier series expansion?

    Homework Statement Which of the signals is not the result of Fourier series expansion? options : (a) 2cos(t) + 3 cos(3t) (b) 2cos(\pit) + 7cos(t) (c) cos(t) + 0.5 Homework Equations Dirichlet conditionsThe Attempt at a Solution From observation, I thought all are periodic and so must be...
  6. F

    MHB Taylor Series Expansion of $e^{-(q+s)^2}$

    Use taylor series to show that the infinite series from n=0 of $$\frac{s^n}{n!}\frac{d^n}{dq^n}(e^{-q^2})=e^{-(q+s)^2}$$
  7. Y

    Help in deriving series expansion of Helmholtz Equation.

    Orthogonality of spherical bessel functions Homework Statement Proof of orthogonality of spherical bessel functions The book gave \int_{0}^{a}\int_{0}^{2\pi}\int_{0}^{\pi} j_{n} (\lambda_{n,j}r) j_{n'} (\lambda_{n',j'}r) Y_{n,m}(\theta,\phi)\overline{Y}_{n',m'}(\theta,\phi) \sin\theta...
  8. M

    Expanding Inhomogeneous Poisson Processes Using Taylor Series

    I'm at the end of a very long Poisson Processes question, involving inhomogeneous Poisson Processes. I just need to be able to expand the following expression to be able to complete the question. exp[{(sin ∏h)/∏} -h] Would anyone please be able to provide some help, with steps please!
  9. J

    Please identify a series expansion

    Hi, Could someone please identify the following series expansion for me, if possible: $$f(x) = 1/h\int_x^{x+h}f(t)dt + A_{1}Δf(x) + A_{2}hΔf'(x) +...+ A_{m-1}h^{m-2}Δf^{(m-2)}(x) +r$$ where $$A_{m}$$ are, as far as I know, plain constants and $$Δ = [x, x+h]$$. I think this result was...
  10. T

    Series Expansion: Proving ln(1+x)/x

    Homework Statement Show that 1- x/2 + x^2/3 - x^3/4 + x^4/5... (-1)^n (x^n)/(n+1) = ln(1+x)/x with |x| < 1 Homework Equations The Attempt at a Solution finding derivative of the function multiplied by x d/dx(xS(x)) = 1 -x + x^2 - x^3 + x^4 - x^5 +... absolute value of this...
  11. U

    Laurent Expansion for 1/(z^2-1) in the Annulus 1 < |z-2| < 3

    Homework Statement Find the Laurent expansion for \frac{1}{z^2-1} in the annulus 1 < |z-2| < 3 The Attempt at a Solution I've gotten to the last parts but getting stuck there. First I expanded the denominator and did a partial fraction decomposition and arrived at...
  12. I

    Series expansion of an integral at infinity

    Hello, I'm fiddling with Wolfram Alpha and I can't find a definition of what do they mean by the "Series expansion of the integral at x -> inf". In particular, I have two divergent integrals and I am wondering whether their ratio is some finite number. Here it is: \left[\int_0^{\infty}...
  13. M

    Laurent Series Expansion of Electrostatic Potential

    Homework Statement Consider a series of three charges arranged in a line along the z-axis, charges +Q at z = D and charge -2Q at z = 0. (a) Find the electrostatic potential at a point P in the x, y-plane at a distance r from the center of the quadrupole. (b) Assume r >> D. Find the...
  14. J

    How to Expand Finite Series in a Shorter Form?

    Hello :blushing: How to do expand this: (\sum_{j=1}^{n}(X(t_j)-X(t_{j-1}))^2 - t)^2 where X(t_j)-X(t_{j-1}) = \Delta X_j to this: (\sum_{j=1}^{n}(\Delta X_j)^4 + 2*\sum_{i=1}^{n}\sum_{j<i}^{ }(\Delta X_i)^2(\Delta X_j)^2 -2*t*\sum_{j=1}^{n}(\Delta X_j)^2+t^2I get near the North Pole... but...
  15. T

    How Do You Find the Power Series Expansion of \( e^z \) at \( \pi i \)?

    "[F]ind the power-series expansion about the given point for each of the functions; find the largest disc in which the series is valid. 10. ##e^{z}## about ##z_{o} = \pi i##" (Complex Variables, 2nd edition; Stephen D. Fisher, pg. 133)$$f(z) = e^{z} = e^{z-a} \cdot e^{a} = e^{a} \cdot \sum...
  16. E

    Series Expansion of $\frac{1}{n^{k (k + 1)/(2 n)}(2 k n - k (1 + k) \ln n)^2}$

    Hi, I'm studying the asymptotic behavior (n -> infinity) of the following formula, where k is a given constant. \frac{1}{n^{k (k + 1)/(2 n)}(2 k n - k (1 + k) \ln n)^2} I'm trying to do a series expansion on this equation to give the denominator a simpler form so that it is easier to make an...
  17. Y

    Series expansion of Sine Integral Si(x)

    This is not really a homework, I am trying to expand Si(x) into a series.The series expansion of Si(x) is given in articles: Si(x)=\int_0^x \frac{\sin\theta}{\theta}d\theta=\sum_0^{\infty}\frac {(-1)^k x^{2k+1}}{(2k+1)(2k+1)!} This is my work, I just cannot get the right answer: Si(x)=\int_0^x...
  18. FeDeX_LaTeX

    Laurent Series Expansion of \frac{1}{e^z - 1}

    Homework Statement Determine the Laurent series expansion of \frac{1}{e^z - 1} The attempt at a solution I've spotted that \frac{1}{e^z - 1} = \frac{1}{2}\left( \coth{\frac{z}{2}} - 1\right) but I don't know what to do next. WolframAlpha gives the series centred at 0 as...
  19. STEMucator

    Calculating Taylor Series Expansion for ##f(x)##

    Homework Statement Calculate the Taylor series expansion about x=0 as far as the term in ##x^2## for the function : ##f(x) = \frac{x-sinx}{e^{-x} - 1 + ln(x+1)}## when ##x≠0## ##f(x) = 1## when ##x=0## Homework Equations Some common Taylor expansions. The Attempt at a Solution...
  20. F

    Taylor series expansion of Dirac delta

    I'm trying to understand how the algebraic properties of the Dirac delta function might be passed onto the argument of the delta function. One way to go from a function to its argument is to derive a Taylor series expansion of the function in terms of its argument. Then you are dealing with...
  21. T

    On Taylor Series Expansion and Complex Integrals

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  22. M

    Series expansion around a singular point.

    Hi All, I have a problem involving some special functions (Meijer-G functions) that I'd like to approximate. At zero argument their first derivative vanishes, but their second and all higher derivatives vanish. (c.f. f(x)=x^{3/2}). Playing about with some identities from Gradshteyn and...
  23. marellasunny

    Talor series expansion of roots of algebraic equation

    I have a algebraic equation like so: x^2-1-εx=0 the roots are obviously- x=ε/2±√(1+ε^2/4) How can I expand the expression for the roots- as a taylor series? the answer is given as: x(1)=1+ε/2+ε^2/8+O(ε^3) I am assuming the author expanded the root 'x' in terms of ε before hand and...
  24. M

    Exponential fourier series expansion

    Hey, thanks for taking the time to look ay my post (: I have attached a file which shows the question I am stuck on, and my attempt at working it out. My problem is the answer I get, is different to what my Lecturer gets (shown in the attachment). He worked it out a different way to me, he...
  25. D

    Question about Laurent Series Expansion

    Say we have a simple function like f(z)=4z/[(z-1)(z-3)2] I'll use this example to demonstrate my undertanding of the motivation behind and usefulness of Laurent series: if we examine f(z), we see it is analytic except where z = 1 and where z = 3, which means it can expanded in a Taylor...
  26. C

    Taylor series expansion for gravitational potential energy. GMm/r=mgh near the earth

    Using taylor series expansion to prove gravitational potential energy equation, GMm/r=mgh at distances close to the earth. R= radius of the Earth h= height above surface of the Earth m= mass of object M= Mass of the earth U = - GmM/(R + h) = - GmM/R(1+ h/R) = - (GmM/R)(1+ h/R)^-1 do a...
  27. P

    Evaluate a limit using a series expansion

    Homework Statement Use a series expansion to calculate L = \lim_{x\to\ 1}\frac{\sqrt[4]{80+x}-(3+\frac{(x-1)}{108})}{(x-1)^{2}} Homework Equations A function f(x)'s Taylor Series (if it exists) is equal to \sum_{n=0}^{\infty}\frac{f^{(n)}(x)}{n!}\cdot (x-a)^{n} Newton's binomial theorem...
  28. V

    Power Series Expansion: Find Alternatives to Ʃ ((-1)^(i-1))/i

    I posted this in the homework section, but it's not a homework problem. I basically need to know if the series Ʃ ((-1)^(i-1))/i can be represented in other ways (e.g. a Taylor series, but I doubt it). I know it converges to ln2, but I need to know if there's a series like x^2, x^4, ... or...
  29. V

    Finding Alternative Representations for the Convergent Series Ʃ ((-1)^(i-1))/i

    Homework Statement I basically need to know if the series Ʃ ((-1)^(i-1))/i can be represented in other ways (e.g. a Taylor series, but I doubt it). I know it converges to ln2, but I need to know if there's a series like x^2, x^4, ... or something like it that I can represent the series...
  30. R

    Exploring the Series Expansion of Coth(pi) Using Fourier Analysis

    Homework Statement show coth(\pi)=1/\pi (1+2 \sum\inftyn=1 (1/(1+n^2) Homework Equations The Fourier expansion of e^x is Sinh(\pi)/\pi(1+\sum 2(-1)^m/(1+n^2) (cos(mx)-n sin(nx) The Attempt at a Solution I subbed in Coth(Pi)=1+e^-pi/sinh(pi) =1+1/\pi (1+2 \sum\inftyn=1 (1/(1+n^2)...
  31. T

    Finding the power series expansion of this ln

    Homework Statement Find the series expansion of ln(x + sqrt(1+x2)) Homework Equations ln(1+x) = x - x2 /2 + x3/3 - x4/4 + ... The Attempt at a Solution I don't know how to solve this. If it was ln(1+f(x) ) I know I could substitute the x's for f(x) in the ln(1+x) series...
  32. C

    Laurent Series Expansion Centered on z=1

    Homework Statement Find a Laurent Series of f(z)=\frac{1}{(2z-1)(z-3)} about the point z=1 in the annular domain \frac{1}{2}<|z-1|<2. Homework Equations The Attempt at a Solution By partial fraction decomposition...
  33. T

    Finding this function's series expansion

    Homework Statement I'm trying to find the series expansion of ln[x + (1+x2)1/2]. Homework Equations The Attempt at a Solution I managed to find the MacLaurin series expansion by using the definition of MacLaurin series, which means I had to derive the function multiple times...
  34. S

    Taylor series expansion of a power series.

    If f(x) is a power series on S = (a-r, a+r), we should be able to expand f(x) as a taylor series about any point b within S with radius of convergence min(|b-(a-r)|, |b - (a + r)|) Does anyone have a proof of this or a link to a proof? I have seen it proved using complex analysis, but I...
  35. Sudharaka

    MHB Taylor Series Expansion Explanation

    mbeaumont99's question from Math Help Forum, Hi mbeaumont99, One thing you can do is to find the Taylor series expansion of \(f(x)=a^{x}\) and see whether it is \(\displaystyle \sum t_{n}\). The Taylor series for the function \(f \) around a neighborhood \(b\) is...
  36. L

    How Do You Perform the Taylor Series Expansion of e(a+x)2?

    how do you do the taylor series expansion of e(a+x)2
  37. J

    Series Expansion Newton Dark Matter

    Hi, I hope you can help me. I am a physicist but not a researcher and have currently not the time to study many papers about the subject of dark matter. What I would need for deeper thoughts is an expansion in a series of the Newton law for cases where it has been measured that star rotation...
  38. G

    Laurent Series Expansion of $\frac{1}{z^2-1}$

    Homework Statement Calculate the laurent series expansion about he points specified, classify the singularity and sate the region of convergence for. \frac{1}{z^2 - 1} at (i) z=1 (ii) z=-1 (iii)z=0 Homework Equations The Attempt at a Solution \frac{1}{z^2 - 1} =...
  39. S

    Series expansion of a harmonic oscillator

    Homework Statement Use a series expansion ψ=A0x0+A1x1+A2x2+... to determine the three lowest-order wave functions for a harmonic oscillator with spring constant k and mass m, and show that the engergies are the expected values. Homework Equations Series expansion given above Time...
  40. P

    MHB Express $\frac{1}{z+1}$ as a power series about z=1

    express $\frac{1}{z+1}$ as a power series about z=1. My working: I know $\frac{1}{z+1}=1-z+z^2-z^3+...$ but this is macluarin series.
  41. S

    How do I find the Laurent series expansion of e^z/z*(1-z)?

    Homework Statement expansion of e^z/z*(1-z) Homework Equations The Attempt at a Solution
  42. J

    Series expansion of function of two variables

    This was stated in a lecture: " For r < 1 we can make a series expansion of f(r,u) in terms of powers of r where: f(r,u) = \frac{1}{\sqrt{1+r^2-2ru}} = \sum^{\infty}_{n=0}r^nP_n(u) " Where P_n(u) is a function of u (and is actually the Legendre polynomials). This was stated without real...
  43. S

    Finding a power series expansion for a definite integral

    Homework Statement Find a power series expansion about x = 0 for the function f(x) = ^{1}_{0}\int\frac{1 - e^{-sx}}{s} ds Homework Equations The power series expansion for a function comes of the form f(x) = ^{\infty}_{0}\sum a_{k}x^{k} The Attempt at a Solution I've tried...
  44. S

    MHB Power Series Expansion: Solving for 1/1+z at z=-5 with Radius of Convergence

    I'm trying to find the power series expansion of 1/1+z at z=-5 and the radius of convergence.How should I think and solve this problem? I'm looking for a step by step explanation because I want to understand the mechanics behind it.Thank you.
  45. P

    Taylor series expansion for gravitational force

    Homework Statement The magnitude of the gravitational force exerted by the Earth on an object of mass m at the Earth's surface is Fg = G*M*m/ R^2 where M and R are the mass and radius of the Earth. Let's say the object is instead a height y << R above the surface of the Earth. Using a...
  46. D

    MHB How do I expand $f(z)$ into a power series?

    $\displaystyle f(z) = \frac{4 + 3z}{(z + 1)(z + 2)^2}$ How do I find the power series? I know that $\displaystyle\frac{1}{z+1} = \frac{1}{1-(-z)} = \sum_{n}^{\infty}(-z)^n$ and $\displaystyle\frac{1}{(z+2)^2} = \frac{d}{dz} \frac{-1}{z+2} = \frac{d}{dz} \frac{-1}{1 - (-z-1)} =...
  47. P

    How Does the Born Series Expansion Translate to Spatial Representation?

    Hello. I read about the born series in scattering, |\psi> = (1+G_0V+\ldots)|\psi_0> Now when I want to move to spatial representation, the textbook asserts I should get \psi(\vec{r})=\psi_0(\vec{r}) + \int dV' G_0(\vec{r},\vec{r'}) V(\vec{r'})\psi_0(\vec{r'})+\ldots by operating with...
  48. A

    Problem with numerator in a series expansion

    Homework Statement The problem I am having has to do with part (d) in the picture which I have attached. I have managed to get as far as to determine that the coefficients in the series expansion have the recurrence relation shown below in part (2). From this I think that I have been able to...
  49. S

    Does the Taylor series expansion for e^x converge quickly?

    Hello all, My question is in regards to the Taylor series expansion of f(x)=e^x=1+x+x^2/(2!)+x^3/(3!)... I calculated the value of e^(-2) using the first 4 terms, 6 terms, and then the first 8 terms. I then calculated the relative error to compare it to the true value, depcited by my...
  50. N

    When to use fourier integral instead of fourier series expansion

    Suppose, I have a non-periodic signal for which amplitude spectrum is to be obtained . For this why should Fourier integral be used instead of Fourier series expansion . I want to know when to use Fourier inerals and when to use Fourier series . Please let me know all the details .
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