Find the maclaurin polynomials of orders n=0,1,2,3, and 4, and then find the nth Maclaurin polynomials for the function in sigma notation.

cos(pi*x)

http://img382.imageshack.us/img382/5889/48183237ut4.png

This is the answer from the solution mannual

Can anybody tell me how it finds f(k) first? I mean not just for this question because all the solution about maclaurin and taylor, the solution finds f(k) first.

Another question, I do not understand the NB means? Why is it n/2 ?