Your approach looks pretty good so far.
Yes I want you to find abs(sum) = 0.

Your only problem is that you are are looking at the wrong magnitude and accuracy.
Tip: Between 1 and 1000 are already several 100 zeros.

In general if you approach something like this you need to check roughly first and then go more and more detailed in the regions that contain a change of sign or are close to zero. Especially on complex functions it is useful to look for zeros in real and imaginary part separately.