2
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I tried much but was unable to find the answer. $$f(x) = \frac{1}{3} + \frac{1 \cdot 3}{3\cdot 6} + \frac{1\cdot 3\cdot 5}{3\cdot 6\cdot 9} + \frac{1\cdot 3\cdot 5\cdot 7}{3\cdot 6\cdot 9\cdot 12} \ldots \infty$$

We have to find value of $X$. I think its an expansion of something, but don't know which.
Thank you.
By the way,the Answer is $2$.

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  • $\begingroup$ What is $X$ meant to represent? $\endgroup$ – Antonios-Alexandros Robotis Dec 23 '15 at 16:17
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    $\begingroup$ A good start might be to write the general term as $\displaystyle \frac{(2n)!/n!2^n}{n!3^n} = \binom{2n}n (1/6)^n$ $\endgroup$ – Henning Makholm Dec 23 '15 at 16:18
  • $\begingroup$ By the way, the answer should be $\sqrt{3} - 1$. $\endgroup$ – Sangchul Lee Dec 23 '15 at 16:23
2
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Hint. One may recall the following result, coming from the generalized binomial theorem: $$ \sum_{n=0}^\infty\binom{2n}n x^n=\frac1{\sqrt{1-4x}},\quad |x|<1/4. $$ then take into account @Henning Makholm's comment.

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  • $\begingroup$ @Norsky Yo are welcome. Thanks! $\endgroup$ – Olivier Oloa Dec 24 '15 at 3:13

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