Doing series is fun!

Calculus Level 4

Recall that (1+x)n=1+nx1!+n(n1)x22!+\large (1+x)^n = 1+ \frac{nx}{1!} + \frac{n(n-1)x^2}{2!} + \cdots

for 1<x<1-1<x<1.

Then what is 1+13+1336+135369+?\large 1 + \frac{1}{3} + \frac{1\cdot3}{3\cdot6} + \frac{1\cdot3\cdot5}{3\cdot6\cdot9} + \cdots ?


Notation: !! is the factorial notation. For example, 8!=1×2×3××88! = 1\times2\times3\times\cdots\times8 .

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