Question

Show that
$1+\frac{1}{r}+\frac{1}{r^4}+\frac{1}{r^9}+\frac{1}{r^{16}}+\cdots =1+\frac{1}{r-}\frac{r}{r^3+1-}\frac{r^3}{r^5+1-}\frac{r^5}{r^7+1-}\cdots$.

Answer 1

$\Rightarrow$ We know that,

$\frac{1}{u_1}+\frac{1}{u_2}+\frac{1}{u_3}+.....=\frac{1}{u_1-}\frac{u_1^2}{u_1+u_2}$

It can be proved by

$\frac{1}{u_r}+\frac{1}{u_{r+1}}=\frac{1}{u_r+x_r}$

$x_r=\frac{-u_r^2}{u_r+u_{r+1}}$

Putting $u_1=r,u_2=r^4,u_3=r^9,u_4=r^{16},....$

$\frac{1}{r}+\frac{1}{r^4}+\frac{1}{r^9}+\frac{1}{r^{16}}+....=\frac{1}{r-}\frac{r^2}{r+r^4-}\frac{r^8}{r^4+r^9-}....$

$=\frac{1}{r-}\frac{r}{{1+r}^3-}\frac{r^3}{1+r^5-}\frac{r^5}{r^7+1-}....$

(By simplified reduction)