Let and P_n= S_2 S_3 S_3 S_4 S_4 S_5 S_n S_ n+1

MATHEMATICS · Subjective · AoPS · Mathematics · Sequences & Series · Sequences & Series

Problem

Let

S_n=1+2+3+\dots+n

and

P_n=\frac{S_2}{S_3}\cdot\frac{S_3}{S_4}\cdot\frac{S_4}{S_5}\cdots\frac{S_n}{S_{n+1}},

where

n\in\mathbb{N}

and

n\geq2

. Find

\lim_{n\to\infty}P_n