HMMT 二月 2005 · CALC 赛 · 第 9 题

英文解析

9. Compute
   ∞
   ∑
   .4
   (4 k )!
   k =0
   Solution: e + 1 /e + 2 cos 1
   This is the power series 3
   4 8
   4 x 4 x
   4 + + + · · ·
   4! 8!
   evaluated at x = 1. But this power series can be written as the sum
   ( )
   2 3 4 5 6 7
   x x x x x x x
   1 + + + + + + + + · · ·
   1! 2! 3! 4! 5! 6! 7!
   ( )
   2 3 4 5 6 7
   x x x x x x x + 1 − + − + − + − + · · ·
   1! 2! 3! 4! 5! 6! 7!
   ( )
   2 4 6
   x x x + 2 1 − + − + · · ·
   2! 4! 6!
   x − x = e + e + 2 cos x.
   It follows that the quantity is e + 1 /e + 2 cos 1.
   ′