HMMT 二月 2004 · CALC 赛 · 第 9 题

英文解析

9. Find the positive constant c such that the series
   ∞0
   ∑
   n !
   ( cn )nn =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 + 1)! / ( c ( n + 1)) 1 nnlim = · limn →∞ n →∞nn ! / ( cn ) c n + 1
   is less than one and diverges if it is greater than one. But
   ( ) ( ) − nnn 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]