Sums that have limits involving log can fail to evaluate

[peteroupc]

Unlike with ordinary limits with integers or symbols, certain sums with non-trivial limits that evaluate to an integer, such as those involving base-2 logarithms, can fail to be evaluated. An example follows:

In [32]: # Limits involving logs  
    ...: mm1=(S(1)/5)-Sum((S(1)/2)*(S(1)/(8*2**k-4)),(k,0,log(n,2)-1))  
    ...: # -- Powers of 2  
    ...: pprint([mm1.subs(n,2**i).doit() for i in range(2,5)])  
    ...: # -- n() produces wrong results here; see also #20937  
    ...: pprint([mm1.subs(n,2**i).n() for i in range(2,5)])  
    ...: # Ordinary integer limits  
    ...: mm1=(S(1)/5)-Sum((S(1)/2)*(S(1)/(8*2**k-4)),(k,0,n))  
    ...: pprint([mm1.subs(n,i).doit() for i in range(4,10)])  
    ...: pprint([mm1.subs(n,i).n() for i in range(4,10)])  
    ...:                                                                                                                
⎡         log(4)                        log(8)                        log(16)             ⎤
⎢    -1 + ──────                   -1 + ──────                   -1 + ───────             ⎥
⎢         log(2)                        log(2)                         log(2)             ⎥
⎢        ____                          ____                          ____                 ⎥
⎢        ╲                             ╲                             ╲                    ⎥
⎢         ╲           1                 ╲           1                 ╲            1      ⎥
⎢1         ╲     ────────────  1         ╲     ────────────  1         ╲      ────────────⎥
⎢─ -       ╱       ⎛   k    ⎞, ─ -       ╱       ⎛   k    ⎞, ─ -       ╱        ⎛   k    ⎞⎥
⎢5        ╱      2⋅⎝8⋅2  - 4⎠  5        ╱      2⋅⎝8⋅2  - 4⎠  5        ╱       2⋅⎝8⋅2  - 4⎠⎥
⎢        ╱                             ╱                             ╱                    ⎥
⎢        ‾‾‾‾                          ‾‾‾‾                          ‾‾‾‾                 ⎥
⎣       k = 0                         k = 0                         k = 0                 ⎦
[-0.00083689405191147, -0.00083689405191147, -0.00083689405191147]
⎡ 27    11     1411    -5869    -1459235    -4840333  ⎤
⎢────, ────, ───────, ────────, ──────────, ──────────⎥
⎣8680  9765  9921240  16866108  2462451768  6771742362⎦
[0.00311059907834101, 0.00112647209421403, 0.000142220125710093, -0.00034797595272128, -0.000592594348024607, -0.00071
4783986343277]

[oscargus]

This seems to be solved by #21911 when that is finished.

In [9]: pprint([mm1.subs(n,2**i).doit() for i in range(2,5)])
⎡       13       ⎤
⎢1/30, ───, 1/140⎥
⎣      840       ⎦