TITLE: Summation of this converging series
QUESTION [2 upvotes]: What's the sum of $$1+\frac{1}{3}.\frac{1}{2}+\frac{2}{3}\frac{5}{6}\frac{1}{2^2}+\frac{1\cdot2\cdot5\cdot8}{3\cdot6\cdot9\cdot2^3}+\cdots$$  I think it's the expansion of some expression but can't figure it out. The answer is $\approx 4^{1/3}$ it isnt duplicate as your question is product of consecutive odd terms but mine is differenet with $2$ and difference $3$

REPLY [1 votes]: HINT...consider the binomial expansion of $(1-\frac 12)^{-\frac 23}$
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