Opened 10 years ago
Closed 7 years ago
#11894 closed defect (fixed)
problems with infinite sum
Reported by: | tmonteil | Owned by: | burcin |
---|---|---|---|
Priority: | trivial | Milestone: | sage-6.3 |
Component: | calculus | Keywords: | infinite sum, maxima |
Cc: | tmonteil | Merged in: | |
Authors: | Peter Bruin | Reviewers: | Karl-Dieter Crisman |
Report Upstream: | Fixed upstream, in a later stable release. | Work issues: | |
Branch: | 1dd0f05 (Commits, GitHub, GitLab) | Commit: | 1dd0f05a2421b2ecda14066a0c1dbe4b5bd6f38e |
Dependencies: | #13973, #13712 | Stopgaps: |
Description
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
Change History (9)
comment:1 Changed 10 years ago by
comment:2 Changed 8 years ago by
- Milestone changed from sage-5.11 to sage-5.12
comment:3 Changed 7 years ago by
- Milestone changed from sage-6.1 to sage-6.2
comment:4 Changed 7 years ago by
- Milestone changed from sage-6.2 to sage-6.3
comment:5 Changed 7 years ago by
- Dependencies set to #13973
- Report Upstream changed from N/A to Fixed upstream, in a later stable release.
The bug in item 4 is fixed upstream and after #13973 the code does correctly raise an error:
sage: sum(1/(m^4 + 2*m^3 + 3*m^2 + 2*m)^2, m, 0, infinity) #0: simp_gen_harmonic_number(exp__=1,x__=-1) #1: ratfun_to_psi(ratfun=1/(m^8+4*m^7+10*m^6+16*m^5+17*m^4+12*m^3+4*m^2),var=m,lo=0,hi=inf) #2: simplify_sum(expr='sum(1/(m^4+2*m^3+3*m^2+2*m)^2,m,0,inf)) ... RuntimeError: ECL says: Error executing code in Maxima: Zero to negative power computed.
comment:6 follow-up: ↓ 7 Changed 7 years ago by
- Branch set to u/pbruin/11894-maxima_sum_zero_division
- Commit set to 1dd0f05a2421b2ecda14066a0c1dbe4b5bd6f38e
- Dependencies changed from #13973 to #13973, #13712
- Priority changed from major to trivial
- Status changed from new to needs_review
Here is a doctest. The dependence on #13712 is because the test is inserted directly after the one there.
Points 2 and 3 have in my opinion been answered in comment:1. Point 1 (the result could be simplified more nicely) is something that should be done in Maxima (simplify certain sums of two polygamma functions to trigonometric functions), so I think it shouldn't be an obstacle to closing this ticket.
comment:7 in reply to: ↑ 6 Changed 7 years ago by
Points 2 and 3 have in my opinion been answered in comment:1. Point 1 (the result could be simplified more nicely) is something that should be done in Maxima (simplify certain sums of two polygamma functions to trigonometric functions), so I think it shouldn't be an obstacle to closing this ticket.
Yes, that was essentially my point then. In principle that could be another ticket but I'm not worried about it.
comment:8 Changed 7 years ago by
- Reviewers set to Karl-Dieter Crisman
- Status changed from needs_review to positive_review
comment:9 Changed 7 years ago by
- Branch changed from u/pbruin/11894-maxima_sum_zero_division to 1dd0f05a2421b2ecda14066a0c1dbe4b5bd6f38e
- Resolution set to fixed
- Status changed from positive_review to closed
Hmm, you have quite a few things here. But which of these are a bug, or should be the main focus of this report?
False
just means "can't prove it's True". For this complicated of an expression, it would be very difficult forbool
to prove this. Again, could be enhanced, but not a bug. You may wish to see if some of the Maxima simplifications could help with this?RR
does not take the real part of a number. That said, we should have something that checks this, I think, unless there is an arcane reason (in this huge expression) we can't.