I've been reading the proof the Central limit theorem from Rick Durret's book. In Theorem 3.3.8 he proves the following
If $E[X^2]<\infty $ then $$\tag{1}\phi(t)=1+itE[X]-\frac{t^2}{2}E[X^2]+o(t^2)$$ where $\phi$ denotes the characteristic function of $X$ and $\lim_{t\to 0}o(t^2)/t^2=0$.
For the proof of this theorem he uses the estimate $$\tag{2}\biggl|e^{ix}-\sum_{k=0}^{n}\frac{(ix)^k}{k!}\biggr|\leq \min\biggl\{\frac{2|x|^n}{n!},\frac{|x|^{n+1}}{(n+1)!}\biggr\}$$
Although, just before the proof of this estimate he makes the observation that if $X$ has finite $n-$th moment $E[|X|^n]<\infty$ then $\phi$ is $n$ times differentiable at $0$ and $\phi^{(n)}(0)=E[(iX)^n]$. So, it follows from the Taylor expansion of $\phi$ at $0$ that $$\tag{3}\phi(t)=\sum_{k=0}^{n}\frac{E(iX)^k}{k!}t^k+o(t^n)$$
My question is on the proof of Theorem 3.3.8 why do we need the estimate of the difference of $e^{ix}$ and it's partial sums since before we have proved $(3)$ which gives us the result for $n=2$. After the proof of Theorem 3.3.8 he makes a comment that roughly says "because we dont have to assume that $E[|X|^3]<\infty$", but i cannot see why. Anyone knows why do we have to prove $(2)$ and use it to prove $(1)$ rather than use $(3)$?
Thanks in advance!