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Cho E = 1/31+1/32+1/33+...+1/60
So sánh E với 4/5
Các bác nào làm được thì giúp em với mai em thi rùi.
E = (1/31 +1/32+ 1/33 +1/34+ 1/35 +1/36+ 1/37 +1/38 + 1/39 +1/40) +
( 1/41 +1/42+ 1/43 +1/44+ 1/45 +1/46+ 1/47 +1/48 + 1/49 +1/50)+
(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)
Mà (1/31 +1/32+ 1/33 +1/34+ 1/35 +1/36+ 1/37 +1/38 + 1/39 +1/40) < ( 1/31 . 10) = 1/ 3 ( 10 số hạng)
Tương tự :( 1/41 +1/42+ 1/43 +1/44+ 1/45 +1/46+ 1/47 +1/48 + 1/49 +1/50)<1/4 và
(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)< 1/5
(1/3 + 1/4 + 1/5 ) < 4/5 ( dpcm)
Ưu ái lắm nha!
S = (1/31+1/32+1/33+...+1/40) + (1/41 + 1/42 + ...+ 1/50) + (1/51 + 1/52+...+1/59+1/60)
Mà : (1/31+1/32+1/33+...+1/40) > 1/40 x 10 = 1/4 (gồm 10 số hạng)
Tương tự : (1/41 + 1/42 + ...+ 1/50) > 1/5 ; (1/51 + 1/52+...+1/59+1/60) > 1/6
S > 1/4 + 1/5 + 1/6.
Trong khi đó (1/4 + 1/5 + 1/6) > 3/5
=>S > 3/5 (1)
S = (1/31+1/32+1/33+...+1/40) + (1/41 + 1/42 + ...+ 1/50) + (1/51 + 1/52+...+1/59+1/60)
Mà : (1/31+1/32+1/33+...+1/40) < 1/31 x 10 = 10/30 = 1/3 (gồm 10 số hạng)
=> S < 4/5 (2)
Từ (1) và (2) => 3/5 <S<4/5
so sanh 2 vế nha
vế 1 chứng minh S>3/5
ta có:S=1/31+1/32+.......+1/60>10.1/40+10.1/50+10.1/60=1/4+1/5+1/6=37/60>3/5
vậy S>3/5
vế 2 chứng minh S<4/5
ta có:S=1/31+1/32+.....+1/60<10.1/30+10.1/40+10.1/50=1/3+1/4+1/5=47/60<4/5
vậy S<4/5
S=1/31+1/32+1/33+...+1/60
Vì:S<(1/30+1/30+....+1/30)+(1/40+1/40+...+1/40)+(1/50+1/50+...+1/50)\
=>S<10.1/30+10.1/40+10.1/50=47/60<48/60=4/5
=>S<4/5(1)
Vì:S>(1/40+1/40...+1/40)+(1/60+1/60+...+1/60)
=>S>10.1/40+20.1/60=1/4+2/3=11/12>9/12=3/4
=>S>3/4(2)
Từ (1) và (2) =>3/4<S<4/5
A= (1/31 + 1/32+ ...+ 1/40) +(1/41 +1/42 +...+ 1/50) + (1/51 +1/52 +...+1/60)
A>10/40 + 10/50 + 10/60
A> 1/4 + 1/5 + 1/6
Ta thấy 1/4 + 1/6 = 10/24> 10/25 = 2/5
suy ra A > 1/5+2/5 = 3/5 suy ra đccm
\(B=\frac{1}{1.2}+\frac{1}{3.4}+...+\frac{1}{59.60}\)
\(B=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{59}-\frac{1}{60}\)
\(B=\left(1+\frac{1}{3}+...+\frac{1}{59}\right)-\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{60}\right)\)
\(B=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+..+\frac{1}{59}+\frac{1}{60}\right)-2.\left(\frac{1}{2}+\frac{1}{4}+...+\frac{1}{50}\right)\)
\(B=\left(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{49}+\frac{1}{50}\right)-\left(1+\frac{1}{2}+...+\frac{1}{30}\right)\)
\(B=\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}=A\)
\(A=\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{2016.2017}\right):2\)
\(=\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2016}-\frac{1}{2017}\right):2\)
\(=\left(1-\frac{1}{2017}\right):2\)\(< \)\(\frac{1}{2}\) (Do 1 - 1/2017 < 1)
(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)(1/51 +1/52+ 1/53 +1/54+ 1/55 +1/56+ 1/57 +1/58 + 1/59 +1/60)
E = 1/31+1/32+...+1/60
E > 1/40+1/40+...+1/40+1/41+1/42+...+1/60
E > 20/40+1/41+1/42+...+1/60
E > 1/2+1/60+1/60+...+1/60
E > 1/2 + 1/3 = 5/6
Mà 5/6 > 4/5
=> E > 4/5