How do you calculate the partition function for single spin?
Or, what are the possible values of s for a single spin?
Recall our definition of s:  2s=N_up-N_down.
If N_up=1 and N_down=0, what is s?
What is s for N_up=0, N_down=1?

To calculate the partition function for a single spin,
your sum over all possible states is just a sum
over those two values of s.