CTR
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{n^2}\) <1 (n thuộc Z, n>= 2)
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Tính tổng đại số
\(A=\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{2}{3}+\dfrac{1}{4}+\dfrac{2}{4}+\dfrac{3}{4}-\dfrac{1}{5}-\dfrac{2}{5}-\dfrac{3}{5}-\dfrac{4}{5}+...+\dfrac{1}{10}+\dfrac{2}{10}+...+\dfrac{9}{10}\)
\(B=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{2}{3}+\dfrac{1}{4}+\dfrac{2}{4}+\dfrac{3}{4}+...+\dfrac{1}{n}+\dfrac{2}{n}+...+\dfrac{n-1}{n}\)\(\left(n\in Z,n\ge2\right)\)
CTR\(\dfrac{11}{15}< \dfrac{1}{21}+\dfrac{1}{22}+\dfrac{1}{23}+...+\dfrac{1}{59}+\dfrac{1}{60}< \dfrac{3}{2}\)
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Chứng minh: Adfrac{2^3+1}{2^3-1}.dfrac{3^3+1}{3^3-1}.dfrac{4^3+1}{4^3-1}....dfrac{9^3+1}{9^3-1} dfrac{3}{2}
Bdfrac{1}{2!}+dfrac{1}{3!}+dfrac{1}{4!}+....+dfrac{1}{n!} 1
Cdfrac{1}{2!}+dfrac{2}{3!}+dfrac{3}{4!}+....+dfrac{n-1}{n!} 1
Dleft(1-dfrac{2}{6}right)left(1-dfrac{2}{12}right)left(1-dfrac{2}{20}right)....left(1-dfrac{2}{nleft(n+1right)}right)dfrac{1}{3}
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Chứng minh: \(A=\dfrac{2^3+1}{2^3-1}.\dfrac{3^3+1}{3^3-1}.\dfrac{4^3+1}{4^3-1}....\dfrac{9^3+1}{9^3-1}< \dfrac{3}{2}\)
\(B=\dfrac{1}{2!}+\dfrac{1}{3!}+\dfrac{1}{4!}+....+\dfrac{1}{n!}< 1\)
\(C=\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+....+\dfrac{n-1}{n!}< 1\)
D=\(\left(1-\dfrac{2}{6}\right)\left(1-\dfrac{2}{12}\right)\left(1-\dfrac{2}{20}\right)....\left(1-\dfrac{2}{n\left(n+1\right)}\right)>\dfrac{1}{3}\)
Chứng minh:
Adfrac{1}{1.2.3}+dfrac{1}{2.3.4}+dfrac{1}{3.4.5}+...+dfrac{1}{18.19.20} dfrac{1}{4}
Bdfrac{36}{1.3.5}+dfrac{36}{5.7.9}+dfrac{36}{9.11.13}+...+dfrac{36}{25.27.29} 3
Cdfrac{1}{2^2}+dfrac{1}{3^2}+dfrac{1}{4^2}+...+dfrac{1}{n^2}in 1left(nin N,nge2right)
Ddfrac{1}{4^2}+dfrac{1}{6^2}+dfrac{1}{8^2}+...+dfrac{1}{left(2nright)^2} 4left(nin N,nge2right)
Edfrac{2!}{3!}+dfrac{2!}{4!}+dfrac{2!}{5!}+...+dfrac{2!}{n!} 1left(nin N,nge3right)
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Chứng minh:
\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+\dfrac{1}{3.4.5}+...+\dfrac{1}{18.19.20}< \dfrac{1}{4}\)
\(B=\dfrac{36}{1.3.5}+\dfrac{36}{5.7.9}+\dfrac{36}{9.11.13}+...+\dfrac{36}{25.27.29}< 3\)
\(C=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{n^2}\in< 1\left(n\in N,n\ge2\right)\)
\(D=\dfrac{1}{4^2}+\dfrac{1}{6^2}+\dfrac{1}{8^2}+...+\dfrac{1}{\left(2n\right)^2}< 4\left(n\in N,n\ge2\right)\)
\(E=\dfrac{2!}{3!}+\dfrac{2!}{4!}+\dfrac{2!}{5!}+...+\dfrac{2!}{n!}< 1\left(n\in N,n\ge3\right)\)
\(A=\dfrac{1}{1.2.3}+\dfrac{1}{2.3.4}+....+\dfrac{1}{18.19.20}=\dfrac{1}{2}\left(\dfrac{1}{1.2}-\dfrac{1}{2.3}+\dfrac{1}{2.3}-\dfrac{1}{3.4}+...+\dfrac{1}{18.19}-\dfrac{1}{19.20}\right)\\ =\dfrac{1}{2}\left(\dfrac{1}{2}-\dfrac{1}{19.20}\right)\\ =\dfrac{1}{4}-\dfrac{1}{2.19.20}< \dfrac{1}{4}\)
Cái B TT nhé
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+....+\dfrac{1}{n^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{\left(n-1\right)n}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n-1}-\dfrac{1}{n}\\ =1-\dfrac{1}{n}< 1\)
D TT
E mk thấy nó ss ớ
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CTR :
S=\(1+\dfrac{1}{1!}+\dfrac{1}{2!}+......+\dfrac{1}{2001!}< 3\)
\(S=1+\frac{1}{1!}+\frac{1}{2!}+......+\frac{1}{2001!}=1+1+\frac{1}{2!}+\frac{1}{3!}+....+\frac{1}{2001!}< 1+1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{2000.2001}=2+\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.....-\frac{1}{2001}< 2+1=3\Rightarrow S< 3\text{(đpcm)}\)
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Ta có
\(\dfrac{1}{2!}=\dfrac{1}{1.2};\dfrac{1}{3!}=\dfrac{1}{1.2.3};\dfrac{1}{4!}=\dfrac{1}{1.2.3.4}< \dfrac{1}{3.4};.....;\dfrac{1}{2001!}=\dfrac{1}{1.2.3.4.......2000.2001}< \dfrac{1}{2000.2001}\)
Vậy S<1+1+\(\dfrac{1}{1.2}+\dfrac{1}{2.3}+......+\dfrac{1}{2000.2001}\)
=1+1+1-\(\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+.....+\dfrac{1}{2000}-\dfrac{1}{2001}\)
= 3-\(\dfrac{1}{2001}< 3\)
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1 ,CMR với n \(\in N\),n\(\ge2\).Ta có :
\(\dfrac{1}{2^3}+\dfrac{1}{3^3}+\dfrac{1}{4^3}+........+\dfrac{1}{n^3}< \dfrac{1}{4}\)
2 , \(\dfrac{1}{2}+\dfrac{2}{2^2}+\dfrac{3}{2^2}+.......+\dfrac{100}{2^{100}}< 2\)
3, CM : \(\dfrac{1}{n+1}+\dfrac{1}{n+2}+\dfrac{1}{n+3}+....+\dfrac{1}{3n+1}< 2\)
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31 tháng 1 2018 lúc 21:12
1.Cmr , với mọi số tự nhiên n lớn hơn hoặc bằng 1
a) dfrac{1}{2^2}+dfrac{1}{4^2}+dfrac{1}{6^2}+....+dfrac{1}{left(2nright)^2} dfrac{1}{2}
b) dfrac{1}{3^2}+dfrac{1}{5^2}+dfrac{1}{7^2}+....+dfrac{1}{left(2n+1right)^2} dfrac{1}{4}
2.Cmr với mọi số tự nhiên lớn hơn hoặc bằng 2
Adfrac{1}{2^2}+dfrac{1}{3^2}+dfrac{1}{4^2}+dfrac{1}{5^2}+...+dfrac{1}{n^2} dfrac{2}{3}
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1.Cmr , với mọi số tự nhiên n lớn hơn hoặc bằng 1
a) \(\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+....+\dfrac{1}{\left(2n\right)^2}< \dfrac{1}{2}\)
b) \(\dfrac{1}{3^2}+\dfrac{1}{5^2}+\dfrac{1}{7^2}+....+\dfrac{1}{\left(2n+1\right)^2}< \dfrac{1}{4}\)
2.Cmr với mọi số tự nhiên lớn hơn hoặc bằng 2
\(A=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+\dfrac{1}{5^2}+...+\dfrac{1}{n^2}< \dfrac{2}{3}\)
a) Đặt \(A=\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{\left(2n\right)^2}\)
\(A=\dfrac{1}{2^2}\left(1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}\right)\)
Ta có:
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< \dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{\left(n-1\right)n}\)
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{n-1}-\dfrac{1}{n}\)
\(\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< 1-\dfrac{1}{n}\)
\(\Rightarrow1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< 1-\dfrac{1}{n}+1\)
\(\Rightarrow1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}< 2-\dfrac{1}{n}\)
\(\Rightarrow\dfrac{1}{2^2}\left(1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{n^2}\right)< \dfrac{1}{2^2}\left(2-\dfrac{1}{2}\right)\)
\(\Rightarrow A< \dfrac{1}{2^2}.2-\dfrac{1}{2^2}.\dfrac{1}{2}\)
\(\Rightarrow A< \dfrac{1}{2}-\dfrac{1}{2^3}< \dfrac{1}{2}\)
Vậy \(A< \dfrac{1}{2}\left(Đpcm\right)\)
b) Đặt \(B=\dfrac{1}{3^2}+\dfrac{1}{5^2}+\dfrac{1}{7^2}+...+\dfrac{1}{\left(2n+1\right)^2}\)
Ta có:
\(B< \dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{\left(2n-1\right)\left(2n+1\right)}\)
\(B< \dfrac{1}{2}\left(\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{\left(2n-1\right)\left(2n+1\right)}\right)\)
\(B< \dfrac{1}{2}\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n+1}\right)\)
\(B< \dfrac{1}{2}\left(1-\dfrac{1}{2n+1}\right)\)
\(B< \dfrac{1}{2}\left(\dfrac{2n+1}{2n+1}-\dfrac{1}{2n+1}\right)\)
\(B< \dfrac{1}{2}.\dfrac{2n}{2n+1}\)
\(B< \dfrac{2n}{4n+2}\)
\(B< \dfrac{2n}{2\left(2n+1\right)}\)
\(B< \dfrac{n}{2n+1}\)
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Rút gọn các biểu thức sau:
a) A 1 + dfrac{1}{3} + dfrac{1}{3^2} + dfrac{1}{3^3} +...+ dfrac{1}{3^n}
b) B dfrac{1}{2} - dfrac{1}{2^2} + dfrac{1}{2^3} - dfrac{1}{2^4} +...+ dfrac{1}{2^{99}} - dfrac{1}{2^{100}}
c) C dfrac{3}{2^2} x dfrac{8}{3^2} x dfrac{15}{4^2} ... dfrac{899}{30^2}
(Mình cần gấp ạ)
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Rút gọn các biểu thức sau:
a) A = 1 + \(\dfrac{1}{3}\) + \(\dfrac{1}{3^2}\) + \(\dfrac{1}{3^3}\) +...+ \(\dfrac{1}{3^n}\)
b) B = \(\dfrac{1}{2}\) - \(\dfrac{1}{2^2}\) + \(\dfrac{1}{2^3}\) - \(\dfrac{1}{2^4}\) +...+ \(\dfrac{1}{2^{99}}\) - \(\dfrac{1}{2^{100}}\)
c) C = \(\dfrac{3}{2^2}\) x \(\dfrac{8}{3^2}\) x \(\dfrac{15}{4^2}\) ... \(\dfrac{899}{30^2}\)
(Mình cần gấp ạ)
b, B = \(\dfrac{1}{2}\) - \(\dfrac{1}{2^2}\) + \(\dfrac{1}{2^3}\) - \(\dfrac{1}{2^4}\)+.....+ \(\dfrac{1}{2^{99}}\) - \(\dfrac{1}{2^{100}}\)
2 \(\times\) B = 1 + \(\dfrac{1}{2}\) + \(\dfrac{1}{2^2}\) - \(\dfrac{1}{2^3}\) + \(\dfrac{1}{2^4}\)-.......-\(\dfrac{1}{2^{99}}\)
2 \(\times\) B + B = 1 - \(\dfrac{1}{2^{100}}\)
3B = ( 1 - \(\dfrac{1}{2^{100}}\))
B = ( 1 - \(\dfrac{1}{2^{100}}\)) : 3
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A = 1 + \(\dfrac{1}{3}\) + \(\dfrac{1}{3^2}\)+ \(\dfrac{1}{3^3}\)+......+ \(\dfrac{1}{3^{n-1}}\) + \(\dfrac{1}{3^n}\)
A\(\times\) 3 = 3 + 1 + \(\dfrac{1}{3}\) + \(\dfrac{1}{3^2}\) + \(\dfrac{1}{3^2}\)+....+ \(\dfrac{1}{3^{n-1}}\)
A \(\times\) 3 - A = 3 - \(\dfrac{1}{3^n}\)
2A = 3 - \(\dfrac{1}{3^n}\)
A = ( 3 - \(\dfrac{1}{3^n}\)) : 2
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C = \(\dfrac{3}{2^2}\) \(\times\) \(\dfrac{8}{3^2}\) \(\times\) \(\dfrac{15}{4^2}\) \(\times\) ...........\(\times\) \(\dfrac{899}{30^2}\)
C = \(\dfrac{1\times3}{2^2}\) \(\times\) \(\dfrac{2\times4}{3^2}\) \(\times\) \(\dfrac{3\times5}{4^2}\) \(\times\)........\(\times\) \(\dfrac{29\times31}{30^2}\)
C = \(\dfrac{1\times2\times\left(3\times4\times5\times....\times29\right)^2\times30\times31}{2^2\times\left(3\times4\times5\times.......\times29\right)^2\times30^2}\)
C = \(\dfrac{2\times\left(3\times4\times5\times.....\times29\right)^2\times30}{2\times\left(3\times4\times5\times.....\times29\right)^2\times30}\) \(\times\) \(\dfrac{1\times31}{2\times30}\)
C = 1 \(\times\) \(\dfrac{31}{60}\)
C = \(\dfrac{31}{60}\)
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1. Cho Ndfrac{1}{31}+dfrac{1}{32}+...+dfrac{1}{60}CMR dfrac{3}{5} N dfrac{4}{5}2. Cho Mdfrac{1}{3}-dfrac{2}{3^2}+dfrac{3}{3^3}-dfrac{4}{3^4}+...+dfrac{29}{3^{29}}-dfrac{30}{3^{30}}CMR M dfrac{3}{16}3. Cho Qdfrac{2}{3}+dfrac{8}{9}+dfrac{26}{27}+...+dfrac{3^{2021}-1}{3^{2021}}CMR Qdfrac{4041}{2}
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1. Cho N=\(\dfrac{1}{31}+\dfrac{1}{32}+...+\dfrac{1}{60}\)
CMR \(\dfrac{3}{5}< N< \dfrac{4}{5}\)
2. Cho M=\(\dfrac{1}{3}-\dfrac{2}{3^2}+\dfrac{3}{3^3}-\dfrac{4}{3^4}+...+\dfrac{29}{3^{29}}-\dfrac{30}{3^{30}}\)
CMR \(M< \dfrac{3}{16}\)
3. Cho Q=\(\dfrac{2}{3}+\dfrac{8}{9}+\dfrac{26}{27}+...+\dfrac{3^{2021}-1}{3^{2021}}\)
CMR \(Q>\dfrac{4041}{2}\)