My question is about the proof of central limit theorem (CLT) in probability theory.

In the proof of CLT (see, for example, Kay, Steven. Intuitive probability and random processes using MATLAB®. Springer Science & Business Media, 2006, pages: 513-514), Taylor expansion is used and the so-called remainder terms above the third term (including the third term) of the Taylor expansion are neglected as N goes to infinity because those terms contain powers of N [(3/2)th, 2nd , (5/2)th etc powers] in their denominators, and hence, get small as N goes to infinity. On the other hand, the second term of the Taylor expansion contains N (1st power of N) in its denomiator. By the same argument, shouldn't this term be ignored as well, since it also goes to zero as N goes to infinity ?