HMMT 二月 2008 · CALC 赛 · 第 9 题
HMMT February 2008 — CALC Round — Problem 9
题目详情
- [ 7 ] Evaluate the limit lim n 1 · 2 · · · · · n . n →∞ ∫ 1
解析
- [ 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 2 n n n n n k =1 k =1 ∫ 1 We can recognize this as the as Riemann sum expansion for the integral x ln x dx , and thus the 0 limit of the above sum as n → ∞ equals to the value of this integral. Evaluating this integral using integration by parts, we find that ∫ ∫ ∣ 1 1 1 1 x 1 ∣ 2 x ln x dx = x ln x − dx = − . ∣ 2 0 2 4 0 0 − 1 / 4 Therefore, the original limit is e . 2 ∫ 1