HMMT 二月 2008 · CALC 赛 · 第 9 题

英文解析

9. [ 7 ] Evaluate the limit lim n 1 · 2 · · · · · n .
   n →∞
   − 1 / 4
   Answer: e Taking the logarithm of the expression inside the limit, we find that it is
   ( ) ( )
   n n
   ∑ ∑
   1 1 1 1 k k − 1 + ln n + k ln k = ln .
   2 n n n n n 2
   k =1 k =1
   ∫
   We can recognize this as the as Riemann sum expansion for the integral x ln x dx , and thus the 1
   limit of the above sum as n → ∞ equals to the value of this integral. Evaluating this integral using 0
   integration by parts, we find that
   ∫ ∫
   ∣
   1 1
   1 x 11
   ∣
   x ln x dx = x ln x − dx = − .2
   ∣
   2 0 2 4
   0 0 − 1 / 4
   Therefore, the original limit is e .
   ∫2 1