HMMT 二月 2008 · 代数 · 第 7 题
HMMT February 2008 — Algebra — Problem 7
题目详情
- [ 5 ] Compute . n + k 2 n =1 k =1 ◦ ◦
解析
- [ 5 ] Compute . n + k 2 n =1 k =1 1 4 Answer: We change the order of summation: 9 ∞ n − 1 ∞ ∞ ∞ ∑ ∑ ∑ ∑ ∑ k k 1 k 4 = = = . n + k k n k 2 2 2 4 9 n =1 k =1 k =1 n = k +1 k =1 (The last two steps involve the summation of an infinite geometric series, and what is sometimes called an infinite arithmetico-geometric series. These summations are quite standard, and thus we omit the details here.) ◦ ◦ ◦