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HMMT 二月 1998 · ADV 赛 · 第 3 题

HMMT February 1998 — ADV Round — Problem 3

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. Find the sum of the infinite series 1 + 2 ( ) + 3 ( ) + 4 ( ) + Κ 1998 1998 1998 2 4 2 2 2 sin A ( cos 3 A + cos A + 3 sin A + sin A cos A )
解析
  1. Find the sum of the infinite series 1 1 1 2 3 1 + 2( ) + 3( ) + 4( ) + . . . . 1998 1998 1998 ( ) 2 1998 3992004 Answer: or . We can rewrite the sum as 1997 3988009 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 2 2 3 2 3 1 1 1 1 1 1 1 1 + + + . . . + + + + . . . + + + . . . + . . . . 1998 1998 1998 1998 1998 1998 1998 Evaluating each of the infinite sums gives ( ) ( ) ( ) 2 1 1 2 ( ) 1 1998 1 1 1998 1 1 1998 1998 2
      • . . . = · 1 + + + . . . = · 1 + + ( ) + . . . , 1 1 1 1997 1998 1998 1997 1998 1998 1 − 1 − 1 − 1998 1998 1998 ( ) 2 1998 3992004 which is equal to , or , as desired. 1997 3988009