JEE Advanced Binomial Theorem 2

Calculus Level 5

1+(13×12)+(13×43×122×12!)+(13×43×73×123×13!)+n terms 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 nn, let SnS_n be the value of the above expression

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

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

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

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