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

$$f^{(r)}(x)=(-1)^{r-1} \frac{(2 r-2) !}{2^{2 r-1}(r-1) !}(1+x)^{\frac{1}{2}-r}$$

Evaluate the series

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

[You may use Taylor's Theorem in the form

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

without proof.]