Sum of an arithmetic infinite series

Question

I have to find the sum of the series : $1+ \frac{2}{6} + \frac{2 \times 5}{6 \times 12} + \frac{2 \times 5 \times 8}{6 \times 12 \times 18} + \cdots$..

I find out the $nth$ term to be $u_n = \frac{2 \times 5 \times 8\times \cdots (3n+2)}{6 \times 12 \times 18 \cdots 6n}$.

After finding the $nth$ term I was unable to proceed.

Any insight will be happily accepted.Also looking for any tips or advices for solving this kind of problems.

Thank you.

See https://math.stackexchange.com/questions/746388/calculating-1-frac13-frac1-cdot33-cdot6-frac1-cdot3-cdot53-cdot6-cdot — lab bhattacharjee, Aug 29 '17 at 17:03

score 1 · Accepted Answer · answered Aug 29 '17 at 16:30

1

Note that $$u_n=\left(-\frac{1}{2}\right)^n\,\binom{-2/3}{n}\,.$$

answered Aug 29 '17 at 16:30

Batominovski

49,629

Yes,I noted it..after that ?, I can't get.. – hiren_garai Aug 29 '17 at 16:58
Then, maybe you should take the hint by Lord Shark the Unknown into consideration. Also, why didn't you take a look at the link given by MyGlasses and Did? – Batominovski Aug 29 '17 at 17:00

Sum of an arithmetic infinite series

1 Answers1