CMR : \(\dfrac{1}{2^2}+\dfrac{1}{2^3}+...+\dfrac{1}{n^2}< 1\) , Với n ∈ N , n > 1
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với n số nguyên dương lớn hơn 1
a) cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< 2-\dfrac{1}{n}\)
b)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< \dfrac{5}{3}\)
CMR: \(\dfrac{1}{1\sqrt{2}}+\dfrac{1}{2\sqrt{3}}+\dfrac{1}{3\sqrt{4}}+...+\dfrac{1}{n\sqrt{n+1}}>2\) với n ϵ N*
với số nguyên dương lớn hơn 1
a)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< 2-\dfrac{1}{n}\)
b)cmr \(\dfrac{1}{1^2}+\dfrac{1}{2^2}+...+\dfrac{1}{n^2}< \dfrac{5}{3}\)
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 : \(\dfrac{1}{n+1}+\dfrac{1}{n+2}+\dfrac{1}{n+3}+....+\dfrac{1}{3n+1}< 2\)
Nguyễn Trần Thành ĐạtXuân Tuấn TrịnhHung nguyenHoang HungQuan Ace Legona giúp với
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CMR với mọi số tự nhiên lớn hơn 2 thì :
\(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{2^n-1}>\dfrac{n}{2}\)
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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CMR: với n là số tự nhiên
\(\dfrac{43}{44}< \dfrac{1}{2\sqrt{1}+1\sqrt{2}}+\dfrac{1}{3\sqrt{2}+2\sqrt{3}}+......+\dfrac{1}{\left(n+1\right)\sqrt{n}+n\sqrt{n+1}}< \dfrac{44}{45}\)
CMR:\(\dfrac{1}{2!}+\dfrac{2}{3!}+\dfrac{3}{4!}+...+\dfrac{n-1}{n!}< 1\)
Trong đó n \(\in\)N, n\(\ge\)2
Lời giải:
\(\frac{n-1}{n!}=\frac{n}{n!}-\frac{1}{n!}=\frac{1}{(n-1)!}-\frac{1}{n!}\). Do đó:
\(\text{VT}=\frac{1}{1!}-\frac{1}{2!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-....+\frac{1}{(n-1)!}-\frac{1}{n!}=1-\frac{1}{n!}< 1\)
Ta có đpcm.
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CMR:\(\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+........+\dfrac{1}{\left(2n\right)^2}< \dfrac{1}{2}\)với n thuộc N,n >1
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