RuntimeError with series

[sagetrac-casamayou]

The function series can not give the power series expansion of f(x)=(1+arctan(x))**(1/x) , while taylor succeeds. Note that the function f can be continuously extended at 0.

sage: taylor((1+arctan(x))**(1/x), x, 0, 3)
1/16*x^3*e + 1/8*x^2*e - 1/2*x*e + e
sage: ((1+arctan(x))**(1/x)).series(x==0, 3)
RuntimeError: power::eval(): division by zero

Subsidiary question: why the functions taylor and series perform similar tasks, but differently?

Depends on #11317

Component: calculus

Keywords: series, taylor

Author: Alexei Sheplyakov, Burcin Erocal

Reviewer: Karl-Dieter Crisman

Merged: sage-4.7.1.alpha1

Issue created by migration from https://trac.sagemath.org/ticket/8943

[kcrisman]

comment:2

Looks like this is in Ginac/Pynac. But maybe it makes sense not to have an answer here? After all, the technical definition would imply that f doesn't have a Taylor series there, if it doesn't even exist. Probably Maxima is more lenient about such things.

[burcin]

Author: Alexei Sheplyakov, Burcin Erocal

[burcin]

comment:3

Attachment: trac_8943-series.patch.gz

This was fixed upstream in ginac. The changes will be in the next pynac release. Patch with doctest is attached.

[burcin]

comment:4

New pynac package with the fix is at #11317.

[burcin]

Dependencies: 11317

[kcrisman]

comment:5

This is nice, and the other examples given by the author also did not work before but now do:

sage: (cos(x)^(sin(x)/x)).series(x==0,9)
1 + (-1/2)*x^2 + 1/8*x^4 + (-1/30)*x^6 + 631/120960*x^8 + Order(x^9)
sage: ((1+x)^(1/x)).series(x==0,9)
(e) + (-1/2*e)*x + (11/24*e)*x^2 + (-7/16*e)*x^3 + (2447/5760*e)*x^4 + (-959/2304*e)*x^5 + (238043/580608*e)*x^6 + (-67223/165888*e)*x^7 + (559440199/1393459200*e)*x^8 + Order(x^9)

Also, the new series does correctly approximate the original function near x=0 :)

[kcrisman]

Reviewer: Karl-Dieter Crisman

[jdemeyer]

Changed dependencies from 11317 to #11317

[jdemeyer]

Merged: sage-4.7.1.alpha1