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Questions tagged [series-expansion]

Questions on dealing with series data and constructing power series expansions of functions.

Analytical tools for series expansions in Mathematica include: Series, ComposeSeries, InverseSeries, SeriesData, SeriesCoefficient.

Series Expansions
Making Power Series Expansions
Operations on Power Series

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Why does $\frac{\partial}{\partial x}O\left(\left(\frac{1}{x}\right)^0\right)$ equal $O\left(\left(\frac{1}{x}\right)^0\right)$ in a series expansion?

When taking the derivative of a series expansion around a finite point, the $O(x^n)$ part is differentiated as expected. $O(x^n)$ becomes $O(x^{n-1})$ except $O(x^0)$ which stays $O(x^0)$. When expanding around infinity, things do not work out that…
Friedrich
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Evaluation of a triple sum does not finish in reasonable time

I'm trying to compute the following triple sum, but no result is produced within a reasonable amount of time. What to do? Sum[1/( i j k (i + j + k + 1)), {i, 1, Infinity}, {j, 1, Infinity}, {k, 1, Infinity}] I use Mathematica 8.0.
user 1357113
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Why does Mathematica fail to series expand this simple expression?

I wanted to expand the function $(x+2)^{x+2}$ around $x = -1$, that is, using Series[(x + 2)^(x + 2), {x, -1, 2}] and Mathematica returns the same expression. Why does this happen? The first term of a series expansion is simply $1^1 = 1$. However,…
TSGM
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How to "prepare" expression for Taylor expansion

I find myself regularly in a situation where I have an expression like $$\frac{m^2+M^2}{(m^2-M^2)^2}$$ with the assumption that $M\gg m$ and the need to expand it up to order $\mathcal{O}(M^{-2})$. By hand, I would first rearrange the terms in the…
Sito
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How do I find a series solution for $e^{-\frac{1}{2}f'(x)} \mathrm{cosh}( f(x) ) = ax + b$?

I am trying to approximate a function $f(x)$ satisfying a relation between $f(x)$ and its first derivative. How do I find a series solution for $$e^{-\frac{1}{2}f'(x)} \mathrm{cosh}( f(x) ) = ax + e^{-\frac{1}{2}f'(0)}$$ about $x=0$? Thus far I've…
Quant
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How to erase the O[x] terms after using Series

I have a matrix in which I have used terms like: Series[Sin[(a qy)/2] , {qy, 0, 2}] How to get rid of the O[qz]^3 afterwards ?
tomphy22
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How to do this series expansion with Mathematica

Consider for large integers $n$ the expression $\sin \left(\pi \sqrt{4 n^2+n}\right)$. Since $\sqrt{4 n^2+n}=2 n \sqrt{1 + \frac{1}{4 n}}$ we can use the standard series for the square root and next the standard series for $sin$ to find a series…
Fred Simons
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Series coefficients for an infinite sum?

I'm working through Carl Bender's Mathematical Physics lectures on YouTube (which are great fun), and I'd like Mathematica's help solving terms in the perturbation series. It would be convenient if expressions like SeriesCoefficient[Sum[a[n] b^n,…
Ian
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Taylor series representation as an infinite sum

I want to see the Taylor series representation for arbitrary functions, e.g. $\sin$. With the Series[] command, I can only see the first $n$ terms. Is there the possibility to show the infinite sum representation? Thanks for the help!
McLawrence
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Trying to get a Laurent expansion of a symbolic function

Im trying to find the Laurent expansion of the function $$f(z):=\frac{a-b}{(z-a)(z-b)},\quad\text{for }0<|a|<|b|$$ around $z=0$ in the annulus defined by $A:=\{z\in\Bbb C:|a|<|z|<|b|\}$. What I did by now is trying to write a function for the…
Masacroso
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Successive Series Expansion Bug (?)

I've found what appears to be a bug in MMA related to taking successive series expansions. I'm providing this minimal example and post as other posts didn't appear to address the issue I found. In particular, MMA seems to get confused in keeping…
astronautgravity
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Power series expansion

I am asking Mathematica for the first 5 terms in a power series expansion like this: nn = 5; a = Sum[Binomial[n!, 2] x^n/n!, {n, 0, nn}]; Series[Log[a], {x, 0, nn}] If I ask for the first 4 terms with the identical code except $nn=4$, I get a…
Geoffrey Critzer
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Peculiarities with Series and fractional exponents or bug?

The documentation of the Series[] function states that it can handle "certain expansions involving negative powers, fractional powers, and logarithms." What are the conditions that dictate whether a function falls into this class of "certain…
nSennett
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Is there a function that, given a rational function, will return the general term of its infinite series expansion?

Is here some way to expand a rational function to an infinite sum in Mathematica, i.e., a series? I want the general term of the series. For example, $\dfrac{2}{3(x-1)^3}$
Pablo
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Series expansion for $\frac{x}{1- \frac{1}{x}}$

I would like to expand $\frac{x}{1- \frac{1}{x}}$ as $$\frac{x}{1- \frac{1}{x}} = x \left( 1+ \frac{1}{x} + \frac{1}{x^2} +\frac{1}{x^3} + \cdots \right) = x + 1 + \frac{1}{x} + \frac{1}{x^2} + \cdots $$ However, I tried Series[1/(1 - 1/x),…
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