@vv 

Thanks to all of you, I will use trunc in place of frac.

@Carl Love I understand. I think the period of b_N is 1/2, not 1/2^N. Thanks for your explanation of these functions. It's clear and complete.

As Rouben point it out, my question is why the limit function in Maple do not give the same results of the derivative.

In[1]: maximize(x^2+4, x = -1 .. 2, location)

Out[1]: 8, {[{x = 2}, 6]}

In[2]: maximize(x^3+4, x = -1 .. 2, location)

Out[2]: 12, {[{x = 2}, 12]}