JEE Advanced Binomial Theorem 2

Calculus Level 5

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

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 \(a+b+c+d\)?

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