# Is That Special?

Calculus Level 5

Define $$f_0(x)=\sqrt{1-x^2}$$.

Now, for $$n>0$$ recursively define $$f_n(x)=\frac{1}{2}\left(f_{n-1}(x)+f_{n-1}\left(x-\frac{1}{2^{n-1}}\right)\right)$$.

For $$n>0$$, the domain of $$f_n(x)$$ is $$-1+\displaystyle \sum_{k=0}^{n-1} \frac{1}{2^k}\leq x \leq 1$$

If $$\displaystyle \lim_{n\to \infty} f_n(1)=\frac{\pi^a}{b}$$ for positive integers $$a,b$$, find $$a+b$$.

Helpful hints:

• $$f_2(x)=\dfrac{\dfrac{\sqrt{1-x^2}+\sqrt{1-(x-1)^2}}{2}+\dfrac{\sqrt{1-(x-.5)^2}+\sqrt{1-(x-1.5)^2}}{2}}{2}$$
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