Some definition of n! where n are the negative integers is taken to be infinity. (this is Gamma function for negative integers).
So 1/(-1)! should be zero. And that is what Mathematica gives
But Maple does not like (-1)!, it gives division by zero, I assume because it is not using the above definition using Gamma function. But then how come Maple is able to find correctly this sum
Which is exp(1). The first term above, when n=0 is 1/(-1)! which Maple does not like. Because if we take the first term as 1/infinity=0 then the sum is the same as if we had
And now it is exp(1) with no problem dealing with negative integer factorial.
Question is: Since Maple does not like negative integer factorial, how comes it accepted the above sum, whose first term contain negative integer factorial?
Just wondering how it did the above, that is all. When I solved this by hand, I used 1/(-1)! = 0 to be able to sum it.