<s>[INST] Sum of the series $1+\frac{1\cdot 3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+\cdots$ Decide if the sum of the series
$$1+\frac{1\cdot3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+ \cdots$$
is: (i) $\infty$, (ii) $1$, (iii) $2$, (iv) $4$.
 [/INST] I believe the general term after the first is
$$\frac{(2n+1)! \cdot 4 \cdot 2}{2^n \cdot n! \cdot 2^{n+2} \cdot (n+2)! } \ , $$
with $ \ n \ \ge \ 1 . $
The series then ought to be convergent.  The series is clearly larger than
$$1 \ + \  \sum_{n=0}^{\infty} \frac{1}{2}  \cdot \left( \ \frac{5}{8} \ \right)^n \ , $$
so the sum is bigger than 2, but not infinite.  That leaves 4 among the choices.
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