If f 0 x - x - x - 1 and f n - 1 - f 0 - f n x for n - 0-1-2 - - then f n x is

Given f _ 0 ( x ) = x x+1 f _ n+1 ( x ) = f _ 0 ( x ) . f _ n ( x ) ⇒ f _ 1 ( x ) = f _ 0 ( x ) . f _ 0 ( x ) =( x x+1 #160; ) #160; ...

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Addition of Fractions

If f 0 x ...

Question

If $f_{0} (x) = \frac{x}{(x + 1)}$ and $f_{n + 1} = f_{0} \circ f_{n} (x)$ for $n = 0, 1, 2, \dots$ then $f_{n} (x)$ is

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⟨ f_{0} (x), f_{1} (x), f_{2} (x), \dots ⟩

f_{0} (x) = 1, f_{1} (x) = x, {(\begin{matrix} f_{n} (x) \end{matrix})}^{2} - 1 = f_{n + 1} (x) f_{n - 1} (x)

n \geq 1

f_{n} (x)

f_{n} (x) = e^{f_{n - 1} (x)}

n ϵ N

f_{0} (x) = x

\frac{d}{d x} {f_{n} (x)}

f_{n} (x) = e^{f_{n - 1} (x)}

n ϵ N

f_{0} (x) = x

\frac{d}{d x} {f_{n} (x)}

f^{0} (x) = \frac{1}{1 + x}

f^{0} (x) = 1 + x

\leq

f^{n} (x) = f^{0} (f^{n - 1} (x))

f^{1} (x) f^{2} (x) f^{3} (x) \dots f^{6} (x)

f_{1} (x) = \frac{x}{2} + 10 \forall x \in R

f_{n} (x) = f_{1} (f_{n - 1} (x)) \forall n \geq 2, n \in N

{lim}_{n \to \infty} f_{n} (x)

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