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a; A = 1 + 1/2^2 + 1/3^2 + 1/4^2 +...+ 1/100^2 < 2
1 = 1 = 1
1/2^2 < 1/1.2 = 1/1 - 1/2
1/3^2 < 1/2.3 = 1/2 - 1/3
.......................
1/100^2 < 1/99.100 = 1/99 - 1/100
Cộng vế với vế ta có:
A = 1 + 1 - 1/100
A = 2 - 1/100 < 2 (đpcm)
\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\)
\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(A=1-\frac{1}{100}\)
\(A=\frac{99}{100}< 2\left(đpcm\right)\)
ta có :
\(\frac{1}{2.3}>\frac{1}{3^2}>\frac{1}{4.3};\frac{1}{3.4}>\frac{1}{4^2}>\frac{1}{4.5}....\)
Tương tự ta sẽ có :
\(\frac{1}{2^2}+\frac{1}{2.3}+.+\frac{1}{99.100}>A>\frac{1}{2^2}+\frac{1}{3.4}+..+\frac{1}{100.101}\)
hay ta có :
\(\frac{1}{4}+\frac{1}{2}-\frac{1}{3}+..+\frac{1}{99}-\frac{1}{100}>A>\frac{1}{2^2}+\frac{1}{3}-\frac{1}{4}+..+\frac{1}{100}-\frac{1}{101}\)
hay \(\frac{1}{4}+\frac{1}{2}-\frac{1}{100}>A>\frac{1}{4}+\frac{1}{3}-\frac{1}{101}\)
hay ta có : \(\frac{1}{4}+\frac{1}{2}>A>\frac{1}{4}+\frac{1}{3}-\frac{31}{300}\Leftrightarrow\frac{3}{4}>A>\frac{12}{25}\)
vậy ta có điều phải chứng minh
\(a)\) Ta có :
\(\frac{1}{100}A=\frac{100^{2009}+1}{100^{2009}+100}=\frac{100^{2009}+100}{100^{2009}+100}-\frac{99}{100^{2009}+100}=1-\frac{99}{100^{2009}+100}\)
\(\frac{1}{100}B=\frac{100^{2010}+1}{100^{2010}+100}=\frac{100^{2010}+100}{100^{2010}+100}-\frac{99}{100^{2010}+100}=1-\frac{99}{100^{2010}+100}\)
Vì \(\frac{99}{100^{2009}+100}>\frac{99}{100^{2010}+100}\) nên \(1-\frac{99}{100^{2009}+100}< 1-\frac{99}{100^{2010}+100}\)
Do đó :
\(\frac{1}{100}A< \frac{1}{100}B\)\(\Rightarrow\)\(A< B\)
Vậy \(A< B\)
Chúc bạn học tốt ~
\(a)A=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2^{100}-1}\)
\(\Rightarrow A=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+...+\frac{1}{2^{100}}+\frac{1}{2^{100}-1}\)
\(\Rightarrow A=1+\left(\frac{1}{2}+\frac{1}{3}\right)+\left(\frac{1}{2^2}+...+\frac{1}{7}\right)+\left(\frac{1}{2^3}+...+\frac{1}{15}\right)+...+\left(\frac{1}{2^{99}}+...+\frac{1}{2^{100}-1}\right)\)
\(\Rightarrow A< 1+\frac{1}{2}.2+\frac{1}{4}.4+\frac{1}{8}.8+...+\frac{1}{2^{99}}.2^{99}\)
\(\Rightarrow A< 100\left(đpcm\right)\)
\(b)A=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{2^{100}-1}\)
\(\Rightarrow A=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+...+\frac{1}{2^{100}}+\frac{1}{2^{100}-1}+\frac{1}{2^{100}}-\frac{1}{2^{100}}\)
\(\Rightarrow A=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{2^2}\right)+\left(\frac{1}{5}+\frac{1}{2^3}\right)+...+\left(\frac{1}{2^{99}+1}+...+\frac{1}{2^{100}}\right)-\frac{1}{2^{100}}\)
\(\Rightarrow A>1+\frac{1}{2}+\frac{1}{2^2}.2+\frac{1}{2^3}.2^2+...+\frac{1}{2^{100}}.2^{99}-\frac{1}{2^{100}}\)
\(\Rightarrow A>1+\frac{1}{2}.100-\frac{1}{2^{100}}\)
\(\Rightarrow A>51-\frac{1}{2^{100}}>51-1\)
\(\Rightarrow A>50\left(đpcm\right)\)