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Find the sum of the infinite series: $$\frac{1}{6}+\frac{5}{6\cdot 12}+\frac{5\cdot 8}{6 \cdot 12 \cdot 18}+\frac{5 \cdot 8 \cdot 11}{6 \cdot 12 \cdot 18 \cdot 24}+\dots$$

I am not able to relate this series to any of the famous power series . I am not getting any clues how to start . I wrote the general term but that didn't help too.

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    Have you already tried to write down the sequence? – 2017-01-25
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    No I haven't. , But what is the benefit of that? – 2017-01-25
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    You can see ways to simplify the "unknown" sequence into a known one., for example notice that $6\cdot12\cdot18\cdots(6n) = 6^n\cdot (n!)$ – 2017-01-25
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    It's necessary in order to get the limit. – 2017-01-25

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Hint: The $n$-th term is $$\frac{1}{2}\,\binom{-2/3}{n+1}\,\left(-\frac{1}{2}\right)^{n+1}\,,$$ where $n=0,1,2,\ldots$. After some simplification, you should obtain $\dfrac{1}{\sqrt[3]{2}}-\dfrac12$ as the final answer.