#11894: problems with infinite sum
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Reporter: tmonteil | Owner: burcin
Type: defect | Status: new
Priority: major | Milestone: sage-4.7.2
Component: calculus | Keywords: infinite sum, maxima
Work_issues: | Upstream: N/A
Reviewer: | Author:
Merged: | Dependencies:
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A recent post on the number theory list asked to compute the value of the
infinite sum of `1/(m^4 + 2m^3 + 3m^2 + 2m)^2` for `m` between 1 and
infinity.
[https://listserv.nodak.edu/cgi-
bin/wa.exe?A2=ind1109&L=nmbrthry&T=0&P=1149]
Trying it to sage :
{{{
sage: var('m')
sage: s = sum(1/(m^4 + 2*m^3 + 3*m^2 + 2*m)^2, m, 1, infinity)
sage: s
1/12*pi^2 + 9/196*I*sqrt(7)*psi(1/14*(3*sqrt(7) - 7*I)*sqrt(7)) -
9/196*I*sqrt(7)*psi(1/14*(3*sqrt(7) + 7*I)*sqrt(7)) - 1/28*psi(1,
-1/2*I*sqrt(7) + 3/2) - 1/28*psi(1, 1/2*I*sqrt(7) + 3/2) - 1
}}}
The formula is less elegant than the formulas given by people who answered
using two proprietary sfotwares, but does not seem false. Sage is not able
to regognize it:
{{{
sage: bool(s == (-(19/16) + 1/84 * pi^2 * (7 - 3 * sech((sqrt(7) *
pi)/2)^2) + ( 9 * pi * tanh((sqrt(7) * pi)/2))/(28 * sqrt(7))))
False
sage: bool(s == -19/16 + 1/28*pi^2*tanh(1/2*pi*7^(1/2))^2 +
9/196*7^(1/2)*pi*tanh(1/2*pi*7^(1/2)) + 1/21*pi^2)
False
}}}
It is also not able to take the real part of a real number:
{{{
sage: CC(s)
0.0161011600422853
sage: RR(s)
[...]
TypeError: cannot convert -7*I to real number
}}}
Moreover, if we let `m` start to zero, sage does not provide an error but
a value:
{{{
sage: var('m')
sage: s = sum(1/(m^4 + 2*m^3 + 3*m^2 + 2*m)^2, m, 0, infinity)
sage: s
1/12*pi^2 + 9/196*I*sqrt(7)*psi(1/14*(sqrt(7) - 7*I)*sqrt(7)) -
9/196*I*sqrt(7)*psi(1/14*(sqrt(7) + 7*I)*sqrt(7)) - 1/28*psi(1,
-1/2*I*sqrt(7) + 1/2) - 1/28*psi(1, 1/2*I*sqrt(7) + 1/2)
sage: CC(s)
1.20360116004229
}}}
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/11894>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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