HMMT 二月 2004 · CALC 赛 · 第 9 题
HMMT February 2004 — CALC Round — Problem 9
题目详情
- Find the positive constant c such that the series 0 ∞ ∑ n ! n ( cn ) n =0 converges for c > c and diverges for 0 < c < c . 0 0 3 3 2 1 [1] [ n +1]
解析
- Find the positive constant c such that the series 0 ∞ ∑ n ! n ( cn ) n =0 converges for c > c and diverges for 0 < c < c . 0 0 Solution: 1 /e The ratio test tells us that the series converges if ( ) n +1 n ( n + 1)! / ( c ( n + 1)) 1 n lim = · lim n n →∞ n →∞ n ! / ( cn ) c n + 1 is less than one and diverges if it is greater than one. But ( ) ( ) − n n n 1 1 lim = lim 1 + = . n →∞ n →∞ n + 1 n e Then the limit above is just 1 /ce , so the series converges for c > 1 /e and diverges for 0 < c < 1 /e . 3 1 3 2 [1] [ n +1]