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HMMT 二月 2020 · 冲刺赛 · 第 33 题

HMMT February 2020 — Guts Round — Problem 33

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [22] Estimate ∞ ∏ − 1 . 25 n N = n . n =1 An estimate of E > 0 will receive b 22 min( N/E, E/N ) c points.
解析
  1. [22] Estimate ∞ ∏ − 1 . 25 n N = n . n =1 An estimate of E > 0 will receive b 22 min( N/E, E/N ) c points. Proposed by: Sujay Kazi Answer: ≈ 8282580 Solution: We approximate ∞ ∑ ln n ln N = 5 / 4 n n =1 with an integral as ∣ ∞ ∫ ∣ ∞ ( ) ln x ∣ − 1 / 4 − 1 / 4 dx = − 4 x ln x − 16 x ∣ 5 / 4 ∣ x 1 1 = 16 . 16 16 Therefore e is a good approximation. We can estimate e by repeated squaring: e ≈ 2 . 72 2 e ≈ 7 . 4 4 e ≈ 55 8 e ≈ 3000 16 e ≈ 9000000 . 16 16 The true value of e is around 8886111, which is reasonably close to the value of N . Both e and 9000000 would be worth 20 points.