My attempt:
$1+\frac{1}{3}+\frac{1.3}{3.6}+\frac{1.3.5}{3.6.9}+\frac{1.3.5.7}{3.6.9.12}....$
=$1+\frac{1}{6}$.$2\choose 1$+$\frac{1}{6^2}$.$4\choose 2$+$\frac{1}{6^3}$.$8\choose 4$+.......+$\frac{1}{6^{n-1}}$.$2n-2\choose n-1$+...........
But I am perplexed as to what should be my next step?
Options:
(A)$\sqrt{2}$
(B)$\sqrt{3}$
(C)$\sqrt{\dfrac{3}{2}}$
(D)$\sqrt{\dfrac{1}{3}}$
One edit $t_n=\dfrac{1.3.5.7.....(2n-3)}{3^{n-1}(1.2.3....(n-1))}$
$=\dfrac{(2n-2)!}{(6^{n-1}(n-1)!(n-1)!)}$
$=\dfrac{1}{6^{n-1}}$.$2n-2\choose n-1$
Is this going to help me?