#4634 Divergent integral simplifies to -1/4 with simplify_sum

[kcrisman]

In the Sage support list we have the following message about a divergent integral giving a solution with the optional package simplify_sum.

"In maxima 5.47.0 I get:

(%i1) load(simplify_sum)$
(%i2) simplify_sum( sum(n*(-1)^n,n,1,inf));
-1/4

This happens both on sage's ECL based one and on the system maxima (running on SBCL). "

That this is a reversion is seen from the following:

Maxima 5.42.2 http://maxima.sourceforge.net
using Lisp ECL 16.1.2
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) load(simplify_sum)$

(%i2) simplify_sum( sum(n*(-1)^n,n,1,inf));
(%o2)                                 ind

[Barton Willis]

With Maxima 5.48.1, we're back to the incorrect value of ind

(%i11)  simplify_sum(sum(n*(-1)^n,n,1,inf));
(%o11)  ind

This result is wrong and it is due to a bug in the limit

(%i12)  simplify_sum(sum(n*(-1)^n,n,1,N));
(%o12)  ((-1)^N*(2*N+1))/4-1/4

(%i13)  limit(%,N,inf);
(%o13)  ind

The output %o12 is correct, but the output %o13 should be und, not ind.

Thanks for this reporting this bug.

[Barton Willis]

Specifically, this is a bug in simplimplus:

(%i66) limit(((-1)^N*(2*N+1))/4-1/4,N,inf);

1 Call   simplimplus [((-1)^N*N)/2+(-1)^N/4-1/4]
1 Return simplimplus ind
(%o66) ind

[Barton Willis]

For other simplimplus bugs, see #4137 simplimplus bugs.

This bug can be eliminated by altering the bit of simplimplus1 code that returns the limit when one or more terms have an undefined limit:

 (cond (undl
       (cond ((and (or indl zerobl zeroal) 
                   (null infl) (null infinityl)) (return '$und))
              ((or infl minfl infinityl)
                  (setq infinityl (append undl infinityl))); x^2 + x*sin(x)
              (t (return '$und)))) ; 1 + x*sin(x)

[Barton Willis]

Simplified:

(undl
           (cond 
                 ;; When there are inf, minf, or infinity terms, the limit might not be und.
                 ;; For example, limit(x^2+x*sin(x),x,inf). For such cases, append the und 
                 ;; terms to the infinity terms.
                 ((or infl minfl infinityl)
                   (setq infinityl (append undl infinityl)))
                 ;; But when there are no infinity terms, the limit is und; for example
                 ;; limit(1 + x*sin(x),x,inf)
                 (t (return '$und))))

This revision fixes two testsuite bugs

**************** rtest_limit_extra.mac: Problem 271 (line 950) ****************

Input:
limit(sin(x) + x cos(x), x, inf)


Result:
und

... Which was correct, but was expected to be wrong due to a known bug in
 Maxima or SBCL.

**************** rtest_limit_extra.mac: Problem 280 (line 1002) ***************

Input:
       x
limit(3  cos(x) + sin(x), x, inf)


Result:
und

... Which was correct, but was expected to be wrong due to a known bug in
 Maxima or SBCL.

And it fixes the bug in the ticket:

(%i1)  load(simplify_sum)$

(%i2) simplify_sum( sum(n*(-1)^n,n,1,inf));
(%o2)                                 und

[Barton Willis]

Fixed by Commit [bef598]. Tests appended tortest_simplify_sum andrtest_limit_extra. Closing this ticket.