HMMT 二月 2016 · 冲刺赛 · 第 19 题

HMMT February 2016 — Guts Round — Problem 19

[ 11 ] Let ( )( ) 2016 n n ∑ i i +2 i A = lim ( − 1) · ( ) 2 n n →∞ i =0 i +1 1 Find the largest integer less than or equal to . A The following decimal approximation might be useful: 0 . 6931 < ln(2) < 0 . 6932, where ln denotes the natural logarithm function.

解析

[ 11 ] Let ( )( ) n n 2016 ∑ i i +2 i A = lim ( − 1) · ( ) 2 n n →∞ i =0 i +1 1 Find the largest integer less than or equal to . A The following decimal approximation might be useful: 0 . 6931 < ln(2) < 0 . 6932, where ln denotes the natural 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) · , ( ) 2 n ( i + 2)( n − i ) i =0 i =0 i +1 So ( )( ) n n 2016 2016 2016 i ∑ ∑ ∑ ( i + 1) ( − 1) i i +2 i i lim ( − 1) · = ( − 1) · = 1 − ≈ ln(2) . ( ) 2 n n →∞ ( i + 2) i i =0 i =0 i =2 i +1 1 1 Then ≈ ≈ 1 . 44, so the answer is 1. A ln(2)