HMMT 二月 2001 · 团队赛 · 第 9 题
HMMT February 2001 — Team Round — Problem 9
题目详情
- Compute for a > 1. i a i =1
解析
- Compute for a > 1. i a i =1 2 3 Solution: The sum S = a + ax + ax + ax + · · · for x < 1 can be determined by realizing ∞ ∑ a ai 2 3 that xS = ax + ax + ax + · · · and (1 − x ) S = a , so S = . Using this, we have = i 1 − x a i =1 ∞ [ ] [( ) ( ) ] ∑ i 1 2 3 1 1 1 1 1 1 a = a + + + · · · = a + + + · · · + + + + · · · + · · · = i 2 3 2 3 2 3 4 a a a a a a a a a a i =1 [ ] [ ] ( ) 2 1 1 1 1 1 a 1 1 a a + + + · · · = 1 + + + · · · = . 2 2 1 − a a 1 − a a 1 − a 1 − a a a 1 − a