summation of infinite series

Question

Find$$S=1+\frac{1}{6}+\frac{1}{18}+\frac{7}{324}+\cdots\infty$$

I could figure this pattern for denominators: $t_k=3t_{k-1}t_{k-2}$ but not sure how to proceed.

Answer 1

Let me try for Newton's generalised binomial theorem with $x=1$

So, we have $\displaystyle ry=\frac16\ \ \ \ (1)$ and $\displaystyle\frac{r(r-1)}{2!}y^2=\frac1{18}\ \ \ \ (2)$

Squaring $(1)$ and dividing by $(2)$ we get $\displaystyle r=-\frac13$ and consequently $\displaystyle y=-\frac12$

which satisfies $$\frac{r(r-1)(r-2)}{3!}y^3=\frac7{324}$$

So, the sum will be $$\left[1+\left(-\frac12\right)\right]^{-\frac13}$$