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Question Number 4132 by prakash jain last updated on 29/Dec/15

$$\mathrm{Is}\mathrm{there}f\left(n\right)\mathrm{such}\mathrm{that}$$$$\underset{x\to 0}{\lim }\left[\underset{n=1}{\sum ^{\mathrm{\infty }}}f\left(n\right)x^n\right]\ne 0$$$$f\left(n\right)\mathrm{is}\mathrm{independent}\mathrm{of}x.$$

Commented by Yozzii last updated on 29/Dec/15

$$s=\underset{n=1}{\sum ^{\mathrm{\infty }}}f\left(n\right)x^n=f\left(1\right)x+f\left(2\right)x^2+f\left(3\right)x^4+$$$$f\left(4\right)x^4+f\left(5\right)x^5+f\left(6\right)x^6+...$$$$f\left(n\right)=\frac{1}{n-k}k\in \mathbb{N}$$$$\Rightarrow s=...+\frac{x^k}{k-k}+...$$$$sdoesnotexist\mathrm{\forall }x\in \mathbb{C}.$$$$Onecannotthenknowif$$$$l=\underset{x\to 0}{\lim }\left\{\underset{n=1}{\sum ^{\mathrm{\infty }}}f\left(n\right)x^n\right\}=0ornot.$$$$$$

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