In the proof of central limit theorem via moment generating function, there is the following step:
$$M\left(\frac{n}{\sqrt t}\right)= 1+ \sigma^2\frac{t^2}{2n}+O(\frac{1}{n^{3/2}})$$
Is it necessary to assume finite third moment to use taylor's inequality to derive this step? How can it be derived assuming finite second moment?
The proof is on page 6 of this document .