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Question Number 130399 by gowsalya last updated on 25/Jan/21

Answered by Dwaipayan Shikari last updated on 25/Jan/21

$$f\left(x\right)=\underset{n=0}{\sum ^{\mathrm{\infty }}}{a}_n{x}^n=a_0+a_{1}x+a_2{x}^2+a_3{x}^3+...$$$$f\left(0\right)=a_0$$$$f^{\prime }\left(0\right)=a_1$$$$f^{″}\left(0\right)=2a_2$$$$f^{‴}\left(0\right)=6a_3$$$$f^{iv}\left(0\right)=24a_4$$$$..f^n\left(0\right)=n!a_n\Rightarrow \frac{f^n\left(0\right)}{n!}=a_n\Rightarrow \frac{f^n\left(0\right)}{n!}{x}^n=a_n{x}^n$$$$\Rightarrow \underset{n=0}{\sum ^{\mathrm{\infty }}}\frac{f^n\left(0\right)}{n!}{x}^n=\underset{n=0}{\sum ^{\mathrm{\infty }}}{a}_n{x}^n=f\left(x\right)$$

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