$1+\left ( \dfrac{1}{3}\times\dfrac{1}{2} \right ) + \left ( \dfrac{1}{3}\times\dfrac{4}{3}\times\dfrac{1}{2^2}\times\dfrac{1}{2!} \right )+\left ( \dfrac{1}{3}\times\dfrac{4}{3}\times\dfrac{7}{3}\times\dfrac{1}{2^3}\times\dfrac{1}{3!} \right ) +\cdots n \text{ terms}$

For natural numbers $n$, let $S_n$ be the value of the above expression

Suppose that for positive integers $a,b,c,d$ with $\text{gcd}(a,b) = \text{gcd}(c,d) = 1$, we have

$\displaystyle\lim_{n\rightarrow\infty} S_n=\left (\dfrac{a}{b}\right )^\dfrac{c}{d}$

What is the value of $a+b+c+d$?