CHứng minh rằng 1 + \(\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^{1999}}>1000\)
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Cho A = \(\dfrac{1001}{1000^2+1}\)+\(\dfrac{1001}{1000^2+2}\)+\(\dfrac{1001}{1000^2+3}\)+...+\(\dfrac{1001}{1000^2+1000}\)
Chứng minh rằng 1<\(^{A^2}\)<4
Tổng A có 1000 số hạng.
Vậy
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Tổng A có 1000 số hạng
A>(1001/1000^2+1000)*1000=1001*1000/1000*(1000+1)=1
A<(1001/1000^2)*1000=1001/1000=1+1/1000<1
Vậy 1<A<2 nên 1<A^2<4
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\(A=\dfrac{1001}{1000^2+1}+\dfrac{1001}{1000^2+2}+\dfrac{1001}{1000^3+3}+.....+\dfrac{1001}{1000^2+100}\)Chứng minh rằng 1<A2<4
chứng minh rằng
\(\dfrac{1}{1000}+\dfrac{1}{1002}+\dfrac{1}{1004}+...+\dfrac{1}{2000}< \dfrac{1}{2}\)
ủa bạn ơi, lớn hơn 1/2 hay bé hơn 1/2 vậy bạn
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Chứng tỏ :
\(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^{1999}}>1000\)
So sánh :
a) Chứng minh rằng : M = \(\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+.......+\dfrac{1}{100!} \)
Chứng minh rằng : M <1 .
b) Chứng minh rằng : N = \(\dfrac{9}{10!}+\dfrac{9}{11!}+\dfrac{9}{12!}+........+\dfrac{9}{1000!}\)
Chứng minh rằng : N < \(\dfrac{1}{9!}\)
a, Ta có :
\(M=\dfrac{1}{1\cdot2}+\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{1\cdot2\cdot3\cdot4}+...+\dfrac{1}{1\cdot2\cdot3\cdot...\cdot100}\\ < \dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{99\cdot100}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-...+\dfrac{1}{99}-\dfrac{1}{100}\\ =1-\dfrac{1}{100}=\dfrac{99}{100}< 1\\ \Rightarrow M< 1\\ \RightarrowĐpcm\)
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Chứng minh rằng:
a) \(\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+...\dfrac{2018}{2019!}\)<1
b) \(\dfrac{1.2-1}{2!}+\dfrac{2.3-1}{3!}+...+\dfrac{999.1000-1}{1000!}\)<2
a) \(\dfrac{1}{2!}+\dfrac{2}{3!}+...+\dfrac{2018}{2019!}\\ =\left(\dfrac{1}{1!}-\dfrac{1}{2!}\right)+\left(\dfrac{1}{2!}-\dfrac{1}{3!}\right)+...+\left(\dfrac{1}{2018!}-\dfrac{1}{2019!}\right)\\ =1-\dfrac{1}{2019!}< 1\)
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b) \(\dfrac{1\cdot2-1}{2!}+\dfrac{2\cdot3-1}{3!}+...+\dfrac{999\cdot1000-1}{1000!}\\ =\dfrac{1\cdot2}{2!}-\dfrac{1}{2!}+\dfrac{2\cdot3}{3!}-\dfrac{1}{3!}+...+\dfrac{999-1000}{1000!}-\dfrac{1}{1000!}\\ =\dfrac{1}{1!}-\dfrac{1}{2!}+\dfrac{1}{1!}-\dfrac{1}{3!}+\dfrac{1}{2!}-\dfrac{1}{4!}+...+\dfrac{1}{999!}+\dfrac{1}{1000!}\\ =1+1-\dfrac{1}{1000!}\\ =2-\dfrac{1}{1000!}< 2\)
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Chứng minh rằng 1 < A2<4 biết :
\(A=\dfrac{1001}{1000^2+1}+\dfrac{1001}{1000^2+2}+...+\dfrac{1001}{1000^2+1000}\)
Chứng minh rằng :
a) \(\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+...+\dfrac{1}{100!}< 1\)
b) \(\dfrac{9}{10!}+\dfrac{9}{11!}+\dfrac{9}{12!}+...+\dfrac{9}{1000!}< \dfrac{1}{9!}\)
a) Đặt :
\(A=\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+.................+\dfrac{1}{100!}\)
Ta thấy :
\(\dfrac{1}{2!}=\dfrac{1}{1.2}\)
\(\dfrac{1}{3!}=\dfrac{1}{1.2.3}\)
\(\dfrac{1}{4!}=\dfrac{1}{1.2.3.4}< \dfrac{1}{3.4}\)
.....................................
\(\dfrac{1}{100!}=\dfrac{1}{1.2.3..........100}< \dfrac{1}{99.100}\)
\(\Rightarrow A< \dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...........+\dfrac{1}{99.100}\)
\(A< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...........+\dfrac{1}{99}-\dfrac{1}{100}\)
\(A< 1-\dfrac{1}{100}\)
\(A< \dfrac{99}{100}< 1\)
\(\Rightarrow A< 1\rightarrowđpcm\)
b) Đặt :
\(B=\dfrac{9}{10!}+\dfrac{9}{11!}+\dfrac{9}{12!}+.............+\dfrac{9}{1000!}\)
Ta thấy :
\(\dfrac{9}{10!}=\dfrac{10-1}{10!}=\dfrac{1}{9!}-\dfrac{1}{10!}\)
\(\dfrac{9}{11!}< \dfrac{11-1}{11!}=\dfrac{1}{10!}-\dfrac{1}{11!}\)
...................................................
\(\dfrac{9}{1000!}< \dfrac{1000-1}{1000!}=\dfrac{1}{999!}-\dfrac{1}{1000!}\)
\(\Rightarrow B< \dfrac{1}{9!}-\dfrac{1}{10!}+\dfrac{1}{10!}-\dfrac{1}{11!}+............+\dfrac{1}{999!}-\dfrac{1}{1000!}\)
\(B< \dfrac{1}{9!}-\dfrac{1}{1000!}\)
\(\Rightarrow B< \dfrac{1}{9!}\rightarrowđpcm\)
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Cho số M= 1+\(\dfrac{1}{\sqrt{2}}+\dfrac{1}{\sqrt{3}}+..+\dfrac{1}{\sqrt{10^6}}\)
Chứng minh rằng 1998<M<1999