HMMT 十一月 2009 · 冲刺赛 · 第 33 题
HMMT November 2009 — Guts Round — Problem 33
题目详情
- [ 20 ] Compute
∞
∑
( )1
.
2009 nn =2009
( )
n !n
Note that is defined as .
k k !( n − k )!
2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd
英文原题
[ 20 ] Compute
∞ ∑
n =2009
1
( n
2009
) .
Note that ( n
k
) is defined as n !
k !( n − k )! .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 nd ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUND
解析
英文解析
- [ 20 ] Compute
∞
∑
( )1
.
2009 nn =2009
( )
n !n
Note that is defined as .
k k !( n − k )!
2009
Answer: Observe that
2008
( )
( ) ( )
n n − 1
k + 1 1 1 k + 1−
k k
( )
( ) ( )( ) − =
n − 1 n n n − 1
k kkk k k
( )
n − 1
k + 1
k − 1
( )( )
n n − 1=
k kkk + 1 ( n − 1)! k ! k !( n − k − 1)!( n − k )!
k n !( n − 1)!( k − 1)!( n − k )!=
k + 1 k · k !( n − k − 1)!
k n !=
( k + 1)!( n − k − 1)!
n !=
( )1
n=
k +1
Now apply this with k = 2008 and sum across all n from 2009 to ∞ . We get
∞ ∞
∑ ∑
1 2009 1 1
( ) ( ) ( ) = − .
n nn − 1
2008
2009 2008 2008
n =2009 n =2009
All terms from the sum on the right-hand-side cancel, except for the initial , which is equal to 1,1
2008
( )
2008
∑
∞
1 2009
so we get = .
n =2009 n
2008
( )
2009
2 ANNUAL HARVARD-MIT NOVEMBER TOURNAMENT, 7 NOVEMBER 2009 — GUTS ROUNDnd