곱 수열 극한

$\displaystyle S _{n} = 1 + 2 + \cdots + n = \frac{n ( n + 1 )}{2}$이므로 $\displaystyle S _{n} - 1 = \frac{n ( n + 1 )}{2} - 1 = \frac{n ^{2} + n - 2}{2} = \frac{( n - 1 ) ( n + 2 )}{2}$이다. $\displaystyle \therefore \frac{S _{n}}{S _{n} - 1} = \frac{n ( n + 1 )}{( n - 1 ) ( n + 2 )}$ $\displaystyle \therefore P _{n} = \frac{S _{2}}{S _{2} - 1} \times \frac{S _{3}}{S _{3} - 1} \times \frac{S _{4}}{S _{4} - 1} \times \cdots \times \frac{S _{n}}{S _{n} - 1}$ $\displaystyle = \frac{3 \cdot 2}{1 \cdot 4} \times \frac{4 \cdot 3}{2 \cdot 5} \times \frac{5 \cdot 4}{3 \cdot 6} \times \cdots \times \frac{( n + 1 ) \cdot n}{( n - 1 ) \cdot ( n + 2 )}$ $\displaystyle = \frac{3}{1} \times \frac{n}{n + 2} = \frac{3 n}{n + 2}$ $\displaystyle \therefore \lim\limits _{n \rightarrow \infty} {P _{n}} = \lim\limits _{n\rightarrow \infty {\frac{3 n}{n + 2}}} = 3$