Yes we can have a negative value in sigma.

Can the sum of a series be negative?

That is, the sum of positive numbers to infinity is negative. ... In the same way, it makes no sense to add up the positive numbers to infinity and say it equals -1/12.

What is the rule for summation?

In combinatorics, the rule of sum or addition principle is a basic counting principle. Stated simply, it is the intuitive idea that if we have A number of ways of doing something and B number of ways of doing another thing and we can not do both at the same time, then there are A + B ways to choose one of the actions.