Unusual infinite series

Calculus Level 3

If $$S=1+\dfrac{2}{7}+\dfrac{2\cdot5}{7\cdot9}+\dfrac{2\cdot5\cdot2}{7\cdot9\cdot7}+\dfrac{2\cdot5\cdot2\cdot5}{7\cdot9\cdot7\cdot9}+\dfrac{2\cdot5\cdot2\cdot5\cdot2}{7\cdot9\cdot7\cdot9\cdot7}\cdots$$

And S can be represented by the form $$S=\dfrac{a}{b}$$ for positive coprime integers a,b. Find a+b

This is just a quick filler problem. I'll be comming out with a set of my top $$\approx \cfrac{e^{\pi}-\pi}{2}$$ best problems soon. They are mostly challenging but all have very beautiful answers and computations

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