HMMT 二月 1998 · ADV 赛 · 第 3 题
HMMT February 1998 — ADV Round — Problem 3
题目详情
- 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 )
英文原题
Find the sum of the infinite series 1 2 3 4 1
1998
1
1998
2 1
1998
3
-
-
-
- ( ) ( ) ( ) Κ
-
-
解析
英文解析
- Find the sum of the infinite series
1 1 1
2 3
1 + 2( ) + 3( ) + 4( ) + . . . .
1998 1998 1998
( )
1998 39920042
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
( )
( ) ( )
1 1 22
( )
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
( )
1998 39920042
which is equal to , or , as desired.
1997 3988009