This problem is a little easier that it may look if you can see what the (-1)k is doing. It's basically making the series alternate between + and -. So if k = 0, then (-1)0 = 1, and the number is positive. If k is an odd number, then the number will be negative.
So adding the function values from k = 0 through k = 4 will give
02 - 12 + 22 - 32 + 42 = 0 - 1 + 4 - 9 + 16 = 10.
I'm not totally clear on whether you're supposed to finding partial sums ending at 0, 1, 2, 3, and 4, or just ending at 4. If you need the other partial sums, then you can just stop the series I have above earlier than I stopped it.