Weird value of a sum

[proy87]

Sum(x,(x,1,Integer(3)/2)).doit() gives 15/8. How is it calculated and how sympy handles non-integer bounds?

[oscarbenjamin]

There is an issue #5822 for this.

It's calculated like this:

In [1]: Sum(x, (x, y, z)).doit()                                                                                       
Out[1]: 
   2        2    
  y    y   z    z
- ── + ─ + ── + ─
  2    2   2    2

In [2]: Sum(x, (x, y, z)).doit().subs({y:1, z:Rational(3, 2)})                                                         
Out[2]: 15/8

Some parts of the implementation of Sum assume that the bounds are integers but nothing prevents giving non-integer bounds. There have been suggestions to disallow non-integer bounds and also suggestions to make sense of them in some way.

The behaviour of Sum in a number of ways does not match the definition given in the docstring:
https://docs.sympy.org/latest/modules/concrete.html#sympy.concrete.summations.Sum

Karr's definition referred to there would imply that Sum(f(x), (x, 1, 4)) == f(1) + f(2) + f(3) but that's not what we get:

In [8]: Sum(f(x), (x, 1, 4)).doit()                                                                                    
Out[8]: f(1) + f(2) + f(3) + f(4)

Also Karr's definition would imply that flipping the limits is equivalent to introducing a minus sign but that isn't what we get:

In [17]: Sum(f(x), (x, 4, 1)).doit()                                                                                   
Out[17]: -f(2) - f(3)

So the current behaviour does not match Karr's definition at all as far as I can tell. The behaviour for inverted limits is non-sensical and the behaviour for ordinary limits does not match the documented definition.

Also having looked at Karr's paper there doesn't seem to be any definition of a summation with non-integer bounds. The paper seems to take it as implicitly obvious that the bounds are integers.

It seems that we need to fix what is the actual definition of Sum and then fix the behaviour to make it consistent.