\(f\left(x\right)=\sum\limits^{32}_02^nx^n+2^{33}.x^{33}+\sum\limits^{\infty}_{n=34}2^nx^n=g\left(x\right)+h\left(x\right)+q\left(x\right)\)
\(\Rightarrow f^{\left(33\right)}\left(x\right)=g^{\left(33\right)}\left(x\right)+h^{\left(33\right)}\left(x\right)+q^{\left(33\right)}\left(x\right)\)
- Do \(\forall n< 33\Rightarrow\left(2^nx^n\right)^{\left(33\right)}=0\Rightarrow g^{\left(33\right)}\left(x\right)=0\) ; \(\forall x\)
- Với \(n>33\Rightarrow n-33\ge1\Rightarrow x^{n-33}=0\) khi \(x=0\)
\(\Rightarrow q^{\left(33\right)}\left(0\right)=0\)
- Với \(n=33\Rightarrow h\left(x\right)=2^{33}.x^{33}\Rightarrow h^{\left(33\right)}\left(x\right)=2^{33}.33!\)
\(\Rightarrow f^{\left(33\right)}\left(0\right)=2^{33}.33!\)
