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Question Number 191582 by Matica last updated on 26/Apr/23

$$f\left(x\right)=x^n.find$$$$A=f\left(1\right)+\frac{f^{{}^{\prime }}\left(1\right)}{1}+\frac{f^{{}^{″}}\left(1\right)}{2}+\frac{f^{{}^{‴}}\left(1\right)}{3}+...+\frac{f^{\left(n\right)}\left(1\right)}{n}$$

Answered by aleks041103 last updated on 27/Apr/23

$$f^{\left(k\right)}\left(x\right)={\left(x^n\right)}^{\left(k\right)}=\{\begin{array}{l}\frac{n!}{(n-k)!}{x}^{n-k},k=0,1,...,n\\ 0,k>n\end{array}$$$$\Rightarrow A=1+\underset{k=1}{\sum ^n}\frac{n!}{k(n-k)!}$$

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