<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] Well , imagination is the magic and you will see the best and most elegent method that follows as such:
$$1+\frac{1\cdot3}{6}+\frac{1\cdot3\cdot5}{6\cdot8}+ \cdots=1+\frac{1\cdot3\cdot(6-5)}{6}+\frac{1\cdot3\cdot5\cdot(8-7)}{6\cdot8}+\cdots$$
$$=1+\frac{1\cdot3\cdot6}{6}-\frac{1\cdot3\cdot5}{6}+\frac{1\cdot3\cdot5\cdot8}{6\cdot8}-\frac{1\cdot3\cdot5\cdot7}{6\cdot8}\cdots=1+1\cdot3=1+3=4$$
Well , this telescoping method really helps in majority of cases.:)
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