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PUMaC 2012 · 代数（A 组） · 第 3 题

PUMaC 2012 — Algebra (Division A) — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
PUMaC

题目详情

  1. [ 4 ] Compute ∞ ∑ n + 1 2 2 n ( n + 2) n =1 Your answer in simplest form can be written as a/b , where a, b are relatively-prime positive integers. Find a + b .
解析
  1. [ 4 ] Compute ∞ ∑ n + 1 2 2 n ( n + 2) n =1 Your answer in simplest form can be written as a/b , where a, b are relatively-prime positive integers. Find a + b . Solution: Let K be the result. ∞ ∑ n + 1 K = 2 2 n ( n + 2) n =1 ∞ ∞ 2 ∑ ∑ 1 n + 4 n + 4 4 K + = 2 2 2 ( n + 2) n ( n + 2) n =1 n =1 ∞ ∞ 2 ∑ ∑ 1 1 ( n + 2) 4 K − 1 − + = 2 2 2 4 n n ( n + 2) n =1 n =1 ∞ ∞ ∑ ∑ 5 1 1 4 K − + = 2 2 4 n n n =1 n =1 5 4 K − = 0 4 5 K = 16 So a + b = 5 + 16 = 21. Answer: 21