Doing series is fun!

Calculus Level 4

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

for \(-1<x<1\).

Then what is \[\large 1 + \frac{1}{3} + \frac{1\cdot3}{3\cdot6} + \frac{1\cdot3\cdot5}{3\cdot6\cdot9} + \cdots ? \]

\(\)
Notation: \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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