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It is possible to convert this infinite continued fraction

$\cfrac{1}{-a+\cfrac{b\;f(0)}{a+\cfrac{b\; f(1)}{-a+\cfrac{b\; f(2)}\ddots}}}$

to a special function ? Please, how do it?

where : $(a,b) >0$ and $f(n)=\cfrac{(n+1)^2}{4(n+1)^2-1}$ , $n \in N$, $f(0)=\cfrac{1}{3}$, $f(1)=\cfrac{4}{15}$, $f(2)=\cfrac{9}{35}$, ...

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Maybe you can just take the first $n$ steps of the fraction and set a denominator to $1$ (or the limit of $f$ as $x\to \infty$). Then calculate the limit as $n$ goes to infinity. – Ragnar Dec 20 '13 at 0:16
How do this Please ? – Betatron Dec 21 '13 at 13:54
@Ragnar how to do it? – Shivam Patel Feb 12 '14 at 7:56

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