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\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)
\(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\)
\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(A=1-\frac{1}{100}\)
\(A=\frac{99}{100}< 2\left(đpcm\right)\)
A=1/2+1/2^2+...+1/2^100
2A=1+1/2+...+1/2^99
2A-A=1+[1/2+(-1/2)]+...+(1/2^99-1/2^99)-1/2^100
2A-A=1+0+...+0-1/2^100
A=1-1/2^100
A=1
BÀI 1:
\(P=1+\frac{1}{2}+\frac{1}{3}+........+\frac{1}{2^{100}-1}\)
\(\Leftrightarrow A=1+\frac{1}{2}+\frac{1}{3}+..........+\frac{1}{2^{100}-1}+\frac{1}{2^{100}}-\frac{1}{2^{100}}\)
\(\Leftrightarrow A=1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{2^2}\right)+........+\left(\frac{1}{2^{99}+1}+.......+\frac{1}{2^{100}}\right)-\frac{1}{2^{100}}\)
\(\Leftrightarrow A>1+\frac{1}{2}+\frac{1}{2^2}\cdot2+\frac{1}{2^3}\cdot2^2+........+\frac{1}{2^{100}}\cdot2^{99}-\frac{1}{2^{100}}\)
\(\Leftrightarrow A>1+\frac{1}{2}\cdot100-\frac{1}{2^{100}}\)
\(\Leftrightarrow A>51-\frac{1}{2^{100}}>51-1=50\)
\(\Rightarrow DPCM\)
BÀI 2 :
TA CÓ: \(A=1+\frac{1}{2}+\frac{1}{2^2}+......+\frac{1}{2^{100}}\)VÀ \(B=2\)
= > CẦN CHỨNG MINH \(\frac{1}{2}+\frac{1}{2^2}+.......+\frac{1}{2^{100}}\)NHƯ THẾ NÀO SO VỚI 1
ĐẶT \(C=\frac{1}{2}+\frac{1}{2^2}+.......+\frac{1}{2^{100}}\)
\(\Leftrightarrow2C=1+\frac{1}{2}+.......+\frac{1}{2^{99}}\)
\(\Leftrightarrow2C-C=\left(1+\frac{1}{2}+.....+\frac{1}{2^{99}}\right)-\left(\frac{1}{2}+.....+\frac{1}{2^{100}}\right)\)
\(\Leftrightarrow C=1-\frac{1}{2^{100}}>1\)
\(\Rightarrow A>B\)
b) \(B=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}\)
\(B=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}+\frac{1}{2015}\)
\(B=1-\frac{1}{2015}\)
\(B=\frac{2014}{2015}\)
a) \(A=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot...\cdot\frac{99}{100}\)
\(=\frac{1}{100}\)
b)\(B=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}\)
\(=1-\frac{1}{2015}\)
\(=\frac{2014}{2015}\)
còn lại tự giải nha gần giống như phần b thôi cũng thú vị.
ủng hộ nha