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talolard

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## Homework Statement

I am making a mistake and i can't find it. Pleasde help me

Let [tex] \left\{ a_{n}\right\} [/tex] be the fibonacci sequence and [tex] f(x)=\sum a_{n}x^{n} [/tex]. Prove that [/tex] f(x)=\frac{1}{1-x-s^{2}} [/tex]in its radius of convergence.

Solution

[tex] f(x)=\sum a_{n}x^{n}=1+x+\sum_{n=2}a_{n}x^{n}=1+x+\sum_{n=2}a_{n-1}x^{n}+\sum_{n=2}a_{n-2}x^{n}=1+x+x\sum_{n=2}a_{n-1}x^{n-1}+x^{2}\sum_{n=2}a_{n-2}x^{n-2}=1+x+xf(x)+x^{2}f(x) [/tex]

=[tex] f(x)(x+x^{2})+1+x=f(x)\iff f(x)(1-x-x^{2})=1+x\iff f(x)=\frac{1+x}{(1-x-x^{2})} [/tex]