HMMT 二月 2016 · 冲刺赛 · 第 19 题
HMMT February 2016 — Guts Round — Problem 19
题目详情
英文原题
- [ 11 ] Let
( )( )
2016 n n
∑
i i +2
A = lim ( − 1) ·i
( )
n 2
n →∞
i =0
i +1
Find the largest integer less than or equal to .1
The following decimal approximation might be useful: 0 . 6931 < ln(2) < 0 . 6932, where ln denotes the anatural logarithm function.
解析
英文解析
- [ 11 ] Let
( )( )
n n
2016
∑
i i +2
A = lim ( − 1) ·i
( )
n 2
n →∞
i =0
i +1
Find the largest integer less than or equal to .1
The following decimal approximation might be useful: 0 . 6931 < ln(2) < 0 . 6932, where ln denotes the anatural logarithm function.
Proposed by: Pakawut Jiradilok
Answer: 1
Note
( )( )
2016 n n 2016
∑ ∑
( i + 1)( n − i − 1)
i i +2
i i
( − 1) · = ( − 1) · ,
( )
n 2
( i + 2)( n − i )
i =0 i =0
i +1
( )( )Son n
2016 2016 2016
∑ ∑ ∑i
( i + 1) ( − 1)
i i +2
i ilim ( − 1) · = ( − 1) · = 1 − ≈ ln(2) .
( )
n 2
n →∞
( i + 2) ii =0 i =0 i =2
i +1
1 1
Then ≈ ≈ 1 . 44, so the answer is 1.
A ln(2)