Câu 1: Cho A = \(\frac{1}{2x2}\)+ \(\frac{1}{3x3}\)+\(\frac{1}{4x4}\)+...+\(\frac{1}{2021x2021}\)
a. So sánh A với 1
b. So sánh A với 3\(\frac{3}{4}\)
K
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18 tháng 10 2018
a, \(A=\frac{1}{2\cdot2}+\frac{1}{3\cdot3}+\frac{1}{4\cdot4}+...+\frac{1}{2011\cdot2011}\)
có :
\(\frac{1}{2\cdot2}< \frac{1}{1\cdot2}\)
\(\frac{1}{3\cdot3}< \frac{1}{2\cdot3}\)
\(\frac{1}{4\cdot4}< \frac{1}{3\cdot4}\)
...
\(\frac{1}{2011\cdot2011}< \frac{1}{2010\cdot2011}\)
nên :
\(A< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{2010\cdot2011}\)
\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2010}-\frac{1}{2011}\)
\(\Rightarrow A< 1-\frac{1}{2011}\)
\(\Rightarrow A< \frac{2010}{2011}< 1\)
b, \(A=\frac{2010}{2011}=1-\frac{1}{2011}\)
\(\frac{3}{4}=1-\frac{1}{4}\)
\(\frac{1}{4}>\frac{1}{2011}\)
nên :
\(A>\frac{3}{4}\)
17 tháng 5 2021
\(Giải\)
\(\Rightarrow A=\frac{1}{2}-\frac{1}{2}+\frac{1}{3}-\frac{1}{3}\)\(+\frac{1}{4}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2014}\)
\(A=0+0+0+...+0+0\)
\(\Rightarrow A=0\)
\(a.\)\(A< 1\)
b. \(A< \frac{3}{4}\)
T
2
ND
Nguyễn Đức Trí
VIP
31 tháng 7 2023
\(A=\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+...+\dfrac{1}{2009.2009}\)
\(\dfrac{1}{2.2}< \dfrac{1}{1.2}=1-\dfrac{1}{2}\)
\(\dfrac{1}{3.3}< \dfrac{1}{2.3}=\dfrac{1}{2}-\dfrac{1}{3}\)
\(\dfrac{1}{4.4}< \dfrac{1}{3.4}=\dfrac{1}{3}-\dfrac{1}{4}\)
...
\(\dfrac{1}{2009.2009}< \dfrac{1}{2008.2009}=\dfrac{1}{2008}-\dfrac{1}{2009}\)
\(\Rightarrow A=\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+...+\dfrac{1}{2009.2009}< 1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...\dfrac{1}{2008}-\dfrac{1}{2009}=1-\dfrac{1}{2009}< 1\)
\(\Rightarrow A=\dfrac{1}{2.2}+\dfrac{1}{3.3}+\dfrac{1}{4.4}+...+\dfrac{1}{2009.2009}< 1\)
31 tháng 7 2023
Ta có:
\(\dfrac{1}{2\times2}+\dfrac{1}{3\times3}+\dfrac{1}{4\times4}+...+\dfrac{1}{2009\times2009}< \dfrac{1}{1\times2}+\dfrac{1}{2\times3}+\dfrac{1}{3\times4}+...+\dfrac{1}{2008\times2009}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{2008}-\dfrac{1}{2009}=1-\dfrac{1}{2009}< 1\)
NN
1 tháng 3 2017
\(A>\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}\)
\(A>\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+....+\frac{2015-2014}{2014.2015}\)
\(A>1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}\)
\(A>1-\frac{1}{2015}\)
Mà \(\frac{1}{2015}< \frac{1}{4}\Rightarrow1-\frac{1}{2015}>1-\frac{1}{4}=\frac{3}{4}\Rightarrow A>\frac{3}{4}\)
26 tháng 3 2019
ta co
1/2.2<1/1*2
...
1/2018*2018<1/2017*2018
=>1/2*2+...+1/2018*1018<1/1*2+...+1/2017.2018
.....(tinh 1/1*2+...+1/2017.*2018)
=>1/2*2+...+1/2018*2018<1-1/2018<1
=>1/2*2+...+1/2018*2018<1
PV
0
PT
2
18 tháng 7 2021
\(A=\frac{1}{2.2}+\frac{1}{3.3}+\frac{1}{4.4}+...+\frac{1}{2021.2021}\)
\(=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2021^2}\)
Xét : \(\frac{1}{k^2}\left(k\inℕ^∗\right)\)
\(=\frac{4}{4k^2}< \frac{4}{4k^2-1}=\frac{4}{\left(2k-1\right)\left(2k+1\right)}==2\left(\frac{1}{2k-1}-\frac{1}{2k+1}\right)\)
Áp dụng cho biểu thức A,ta có :
\(A< 2\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{4041}-\frac{1}{4023}\right)\)
\(=2\left(\frac{1}{3}-\frac{1}{4023}\right)=\frac{2}{3}-\frac{2}{4023}< \frac{2}{3}< \frac{3}{4}\)
help me!!!