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HMMT 二月 2005 · CALC 赛 · 第 9 题

HMMT February 2005 — CALC Round — Problem 9

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Compute ∞ ∑ 4 . (4 k )! k =0 ′
解析
  1. Compute ∞ ∑ 4 . (4 k )! k =0 Solution: e + 1 /e + 2 cos 1 3 This is the power series 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. ′