Question

Chứng minh

A=1/2!+2/3!+3/4!+....+2015/2016!<1

B=1/6<M=1/5^2₊1/6²+1/7²+1/100²

C=1/20+1/21+1/22+...+1/200>9/10

 

Answer 1

A = 1/2! + 2/3! + 3/4! + ... + 2015/2016!

A = 2/2! - 1/2! + 3/3! - 1/3! + 4/4! - 1/4! + ... + 2016/2016! - 1/2016!

A = 1 - 1/2! + 1/2! - 1/3! + 1/3! - 1/4! + ... + 1/2015! - 1/2016!

A = 1 - 1/2016! < 1 (đpcm)

M = 1/5² + 1/6² + 1/7² + ... + 1/100²

M > 1/5.6 + 1/6.7 + 1/7.8 + ... + 1/100.101

M > 1/5 - 1/6 + 1/6 - 1/7 + 1/7 - 1/8 + ... + 1/100 - 1/101

M > 1/5 - 1/101 > 1/5 - 1/30 = 1/6 = B

=> M > B (đpcm)

C = 1/20 + 1/21 + 1/22 + ... + 1/200

C > 1/200 + 1/200 + 1/200 + 1/200

       (181 phân số 1/200)

C > 1/200 . 181 = 181/200 > 180/200 = 9/10 (đpcm)