Question: Find the sum of all integer values of c such that $x^2 +cx+\frac{1}{4}c$ has two real, distinct roots.
The discriminant must be greater than zero for a quadratic to have two real, distinct, roots: $$ c^2-(4)\left(\frac{1}{4}\right)c>0\implies c(c-1)>0\implies c \in (-\infty,0) \cup (1,\infty)$$
And now we want the sum: $$...-3-2-1+2+3+4+...$$
Note that we can make pairs $-2+2=0, -3+3=0$ except for $-1$, which is unpaired, and the hence the final answer is $$-1$$
Is the above logic correct? Particularly the part where the infinite series is summed? Is there a flaw in the question?