Question

[Kỳ Tỉ]

chứng minh

a) 1/6<1/5^2+1/6^2+1/7^2+......+1/100^2 < 1/4

b) 4/3<1/11+1/12+....+1/70<5/2

Answer 1

[123456]

a) ta có :1/5^2<1/4.5=1/4-1/5

1/6^2<1/5.6=1/5-1/6

.................

1/100^2<1/99.100=1/99-1/100

=>1/5^2+1/6^2+1/7^2+......+1/100^2 <1/4-1/100=6/25<1/4(1)

ta lại có:1/5^2>1/5.6=1/5-1/6

1/6^2>1/6.7=1/6-1/7

.................

1/100^2>1/100.101=1/100-1/101

=>1/5^2+1/6^2+1/7^2+......+1/100^2>1/5-1/101=96/505>1/6(2)

từ (1)(2) suy ra 1/6<1/5^2+1/6^2+1/7^2+......+1/100^2 < 1/4

Answer 2

[123456]

b)ta có:1/11+1/12+....+1/70=(1/11+1/12+...+1/20)+(1/21+1/22+...+1/30)+(1/31+1/32+...+1/40)+(1/41+1/42+...+1/50)+(1/51+1/52+...+1/60)+(1/61+1/62+...+1/70)>(1/20+1/20+...+1/20)_{(10 phân số 1/20)}+(1/30+1/30+...+1/30)_{(10 phân số 1/30)}+(1/40+1/40+...+1/40)_{(10 phân số 1/40)}+(1/50+1/50+...+1/50)_{(10 phân số 1/50)}+(1/60+1/60+...+1/60)_{(10 phân số 1/60)}=1/2+1/3+1/4+1/5+1/6=29/20>4/3(1)

ta lại có:1/11+1/12+....+1/70=(1/11+1/12+...+1/20)+(1/21+1/22+...+1/30)+(1/31+1/32+...+1/40)+(1/41+1/42+...+1/50)+(1/51+1/52+...+1/60)+(1/61+1/62+...+1/70)<(1/11+1/11+...+1/11)_{(10 phân số 1/11)}+(1/21+1/21+...+1/21)_{(10 phân số 1/21)}+(1/31+1/31+...+1/31)_{(10 phân số 1/31)}+(1/41+1/41+...+1/41)_{(10 phân số 1/41)}+(1/51+1/51+...+1/51)_{(10 phân số 1/51)}+(1/61+1/61+...+1/61)_{(10phân số 1/61)}  =10/11+10/21+10/31+10/41+10/51+10/61=2,311777327<5/2(2)

từ (1)(2)=>4/3<1/11+1/12+....+1/70<5/2