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Ta có \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+\frac{1}{7^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+\frac{1}{5\cdot6}+\frac{1}{7\cdot8}\)
\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-....+\frac{1}{7}-\frac{1}{8}\)
\(\Rightarrow A< 1-\frac{1}{8}< 1\)
Đặt S=1/4+1/16+1/36+...+1/10000
S= 1/4x(1+1/4+1/9+...+1/2500)
S= 1/4x(1+1/2x2+1/3x3+...+1/50x50)
S< 1/4x(1+1/1x2+1/2x3+...1/49x50)
S< 1/4x(1+1-1/2+1/2-1/3+....+1/49-1/50)
S< 1/4x(1+1-1/50)
S< 1/4x(2-1/50)<2/4(2/4=1/2)
S< 1/2
Ta có: \(\frac{1}{4}< \frac{1}{2}\)
\(\frac{1}{16}< \frac{1}{2}\)
... . . .
\(\frac{1}{10000}< \frac{1}{2}\)
\(\frac{1}{10000}+\frac{1}{10000}+...+\frac{1}{10000}< \frac{1}{4}+\frac{1}{6}+\frac{1}{36}+\frac{1}{64}+...+\frac{1}{10000}< \frac{1}{2}+\frac{1}{2}+...+\frac{1}{2}\)(*) (n phân số \(\frac{1}{10000}\) ; n phân số \(\frac{1}{2}\))
Từ đó suy ra \(\frac{1}{4}+\frac{1}{6}+\frac{1}{36}+\frac{1}{64}+...+\frac{1}{1000}< \frac{1}{2}\left(đpcm\right)\)
\(A< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{19\cdot20}\)
\(\Rightarrow A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{99}-\frac{1}{20}\)
\(\Rightarrow A< 1-\frac{1}{20}< 1\)
Đặt \(A=\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+....+\frac{1}{17}\)
Ta có: \(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}+\frac{1}{10}< \frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}+\frac{1}{5}=\frac{6}{5}\left(1\right)\)
\(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+\frac{1}{15}+\frac{1}{16}+\frac{1}{17}< \frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}+\frac{1}{11}=\frac{7}{11}\left(2\right)\)
Từ (1)(2) \(\Rightarrow A< \frac{6}{5}+\frac{7}{11}=\frac{66}{55}+\frac{35}{55}=\frac{101}{55}< \frac{110}{55}=2\)
\(\Rightarrow A< 2\Rightarrow\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+...+\frac{1}{17}< 2\left(đpcm\right)\)
Ta có:
1/5=1/5
1/6<1/5
1/7<1/5
..........
1/10<1/5
=>1/5+1/6+...+1/10<1/5.6=6/5(1)
Lại có :
1/11=1/11
1/12<1/11
1/13<1/11
.............
1/17<1/11
=>1/11+1/12+1/13+...+1/17<1/11.7=7/11(2)
Từ (1)và (2)=>1/5+1/6+...+1/17<6/5+7/11=101/55<110/55=2
=>1/5+1/6+...+1/17<2
ĐPCM
cm \(\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+\frac{1}{10^2}< \frac{1}{2}\)
\(\Leftrightarrow\frac{1}{2^2}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}\right)< \frac{1}{2}\)
\(\Leftrightarrow1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}< 2\)
\(\Leftrightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}< 1\)
ta cần chứng minh điều trên:
Đặt \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}\)
\(A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}\)\(=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+\frac{5-4}{4.5}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}=1-\frac{1}{5}\)
\(\Leftrightarrow A< 1-\frac{1}{5}< 1\)
suy ra đpcm
Đặt biểu thức là A
Ta có:A=\(\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+\frac{1}{64}+\frac{1}{100}\)
\(\Rightarrow\)A=\(\frac{1}{4}+\left(\frac{1}{16}+\frac{1}{36}\right)+\left(\frac{1}{64}+\frac{1}{100}\right)\)
\(\Rightarrow\)A<\(\frac{1}{2}\)
