# Questions tagged [series-expansion]

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

171 questions with no upvoted or accepted answers
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### Bug in SumConvergence

Bug introduced in 10.0.1 and fixed in 12.0.0 Version 11.2.0.0 on MacBook Pro: ...
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### Design considerations behind O (a.k.a. BigOh, a.k.a. Landau Order)

This works without any warnings: O[Log[x]]. This raises a warning: O[x^2]. I have a few questions around this: Why is it a ...
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### How to revert behavior of SeriesData to pre 12.1

In Mathematica 12.0 and earlier, SeriesData[x, 0, {1/u + Log[x/y]}, 0, 3, 1] used to preserve its list of expressions in the form it was given. Now, in ...
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### Getting around Series[Sinc] bug

Bug introduced in 8.0 or earlier and fixed in 10.4 For some reason, Series expansion of Sinc around ...
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### MMA does not provide the correct asymptote for an integral function

Given is the function $$f(x)=\int_0^\infty \mathbb{exp}\left(-\frac{x^2}{2t^2}-t\right)\mathbb{d}t$$ Mathematica returns for the asymptotic behavior $x\to\infty$ using ...
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### Series expansion of Lerch transcendent still buggy?

This series expansion of a Lerch transcendent seems fixed in V12. However, the following still fails: From the definition of a Lerch transcendent, ...
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### How to calculate the series of this function?

I have to calculate the series of the function F[r_] := 1 - a - a*r^(5 + n)/(r^(8 + n) + 1); for r->Infinity for generic ...
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### Another example where using FullSimplify gives different result than Simplify

I believe this question is very similar to Result of Series[expression] is different when I simplify the expression, however, due to my lack of Mathematica experience, I am reluctant to call it a bug. ...
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### Series of LegendreP[a,b,x] takes ages in Mathematica 11

My institution just upgraded to Mathematica 11.3 (from v10) and I'm experiencing a problem that is absent in Mathematica 10. Namely, ...
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### Multivariate FindGeneratingFunction

For one dimensional problems sometimes I'm using the Mathematica function FindGeneratingFunction[seriescoefficient,x] which uses the coefficients of a powerseries ...
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### Change in behaviour of Series from 11.1

In 11.0, Series[(1 - Sqrt[1-4z])/2, {z, 1/4, 1}] gives 1/2 - I Sqrt[z-1/4] + O[z-1/4]^(3/2) as one would expect. From 11.1 ...
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### Series expansion in Infinity issue with Zeta(s) function

With this code: Series[Zeta[s], {s, Infinity, #}] & /@ Range[10] // MatrixForm Series expansion for the Zeta(s) function ...
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### A series in powers of $(a-z)$ instead of $(z-a)$

Sometimes it is more convenient to find a series expansion (e.g., Taylor, Laurent, Puiseux, ...) in powers of $(a-z)$ than in powers of $(z-a)$. For instance, the command ...
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### Quirky behavior of Series[]

This is a "what's going on?" question about MMa behavior, not so much a "how to fix?" This code calculates a Taylor series for a two-term Gaussian Mixture: ...
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### Why does Mathematica not go to specified order in series?

For some reason Mathematica will not evaluate this asymptotic series to the requested order. Inputting: ...
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### Proper treatment of roots and powers in Series?

I have the following problem in Mathematica 9 on Linux. I let Mathematica compute the Series expansion: ...
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### Cannot Understand nth Derivative of x/ArcTan[x]

The nth derivative of x/ArcTan[x]: f[x_, n_] = D[x/ArcTan[x], {x, n}] Evaluates to: I cannot get this general from to return ...
487 views

### Series expansion with fractional exponents

The following series expansion Series[1 + Sum[b[n] (x^(1/4))^n, {n, 1, 3}], {x, 0, 1}] gives terms up to O(x^{5/4}). I would ...
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### Power series raised to powers

Power series raised to powers: I am looking for a way to implement a replacement using this relation explained on wikipedia and in another post here: https://math.stackexchange.com/questions/1471438/...
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### Properties of the prime geometric series

§1. The well known geometric sum is given by g[x_] = Sum[x^n, {n, 0, \[Infinity]}] (* Out[659]= 1/(1 - x) *) Here we consider the "prime geometric series" in ...
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### Negative result of a integral of positive function

Let's try this: I have a function of two arguments, $e$ and $\omega$. First I integrate some function of $(e, \omega)$: ...
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### Error when simplifying a series expression

I have the following function which I call FF[q_,y_,u_] and this function is well known to have a reasonable Taylor expansion in all three variables. For example, ...
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### Unexpected imaginary term in the asymptotic expansion of DawsonF

FullSimplify[Series[2 DawsonF[x], {x, ∞, 8}]] (* -I Exp[-x^2] √π + (1/x + 1/(2 x^3) + 3/(4 x^5) + 15/(8 x^7) + O[1/x]^9) *) What is the reason the term ...
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### Polynomial kernel expansion

I am trying to calculate the polynomial kernel expansion using Mathematica. I have tried the Expand and Simplify functions with ...
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### Discrepancy in the series expansion of $\log(z/(z-1))$ in Mathematica 5

When I calculate the series expansion of $\log\frac{z}{z-1}$ around $0$ in Mathematica 5.2, I obtain: ...
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### Generating function for Newton series?

The function GeneratingFunction gives generating function for Taylor series. Is there a similar function for Newton's series? $$f(x) = \sum_{k=0}^\infty \binom{x}k \Delta^k [f]\left (0\right)$$
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### Expanding a Function in Series works, SeriesCoefficient Doesn't Work

Take the following definitions: ...
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### Changing bounds of summation after differentiating symbolic sums

Suppose, I have a function written as Taylor-Maclaurin series f = Sum[c[n]*x^n, {n, 0, Infinity}] Now, I wish to differentiate this expression with respect to <...
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### Approximation of roots using Series

I am solving a fifth degree polynomial using Series. My equation looks like ...
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### Multivariate Lagrange inversion

I know that the function InverseSeries (Reference here) provides an interface for the Lagrange inversion formula. However I can't find anything on the ...
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### Form a power series from a list

Say I have a list L = {0,1,2,4,6} how can I form a power series with coefficients in L x + 2x^2 + 4x^3 + 6x^4 ?
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### Symbolic perturbation expansion for quantum mechanics using Hellmann-Feynman derivaties

I am interested in some quantum mechanical perturbation expansion for energies. Actually I want to implement these terms $E_n^{(k)}$. As is stated below one can do that using CAS. I would be ...
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### Eliminating higher order trigonometric terms

I am interested in eliminating higher-order trigonometric terms from a long symbolic expression. Specifically I want to reproduce this simplification that is done (in a tutorial I am working through)...
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### Finding symbolic series coefficients

Is there any way to get the symbolic form of a series? For simple series this seems possible e.g. SeriesCoefficent[Exp[x],{x,0,n}] tells me that the general ...
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### Expansion of HypergeometricU

I try to get the first three terms of expansion of HypergeometricU in Mathematica. My input is ...
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### Why do these sums return the same result when the results should be different?

Sum[1, {k, 1, Infinity}, Regularization -> Dirichlet] gives -1/2, which is right. ...
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### Parallelizing / speeding up a replacement rule over large list

I have a list of differentiated expression of the form {f[0,0,0],(f^(a,b,c))[0,0,0],...} where I have left the function as a general ...
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### Formal Power Series and 0^0

I'm having the following problem: I define series = Exp[Sum[Subscript[J, n]/n t^n, {n, 1, \[Infinity]}]] and it's all fine. Also when I ask Mathematica ...
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