Analysis | Part IA, 2004

Let f(x)=(1+x)1/2f(x)=(1+x)^{1 / 2} for x(1,1)x \in(-1,1). Show by induction or otherwise that for every integer r1r \geq 1,

f(r)(x)=(1)r1(2r2)!22r1(r1)!(1+x)12rf^{(r)}(x)=(-1)^{r-1} \frac{(2 r-2) !}{2^{2 r-1}(r-1) !}(1+x)^{\frac{1}{2}-r}

Evaluate the series

r=1(1)r1(2r2)!8rr!(r1)!\sum_{r=1}^{\infty}(-1)^{r-1} \frac{(2 r-2) !}{8^{r} r !(r-1) !}

[You may use Taylor's Theorem in the form

f(x)=f(0)+r=1nf(r)(0)r!xr+0x(xt)nf(n+1)(t)n!dtf(x)=f(0)+\sum_{r=1}^{n} \frac{f^{(r)}(0)}{r !} x^{r}+\int_{0}^{x} \frac{(x-t)^{n} f^{(n+1)}(t)}{n !} d t

without proof.]

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