ta có:

\(\frac{1}{2}-\frac{1}{2}+\frac{1}{3}-\frac{1}{3}+..+\frac{1}{100}-\frac{1}{100}\)

só sánh:

1/2 với 3/4 sẽ có:3/4>1/2

1/3 với 3/4 sẽ có: 3/4>1/3

vì các phân số tiếp theo đều nhỏ hơn 1/2 nên

3/4>A

https://olm.vn/hoi-dap/detail/54671443759.html

Ta có \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+....+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{99.100}\)

             \(=\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+....+\left(\frac{1}{99}-\frac{1}{100}\right)\)

               \(\frac{1}{2}-\frac{1}{100}=\frac{49}{100}< \frac{3}{4}\left(đpcm\right)\)

b) A=\(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^{99}}\)

   3A=\(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{98}}\)

3A-A=\(1-\frac{1}{3^{99}}\)

   2A=\(1-\frac{1}{3^{99}}\)

vì 2A<1

=> A<\(\frac{1}{2}\)

a, \(A=1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{100^2}\)

\(\Rightarrow A< 1+\frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\)

\(\Rightarrow A< 1+\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\right)\)

\(\Rightarrow A< 1+\left(1-\frac{1}{100}\right)\Rightarrow A< 1+1-\frac{1}{100}\Rightarrow A< 2-\frac{1}{100}\Rightarrow A< 2\left(ĐPCM\right)\)

b, \(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2012^2}\)

\(\Rightarrow B< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{2011\cdot2012}\)

\(\Rightarrow B< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2011}-\frac{1}{2012}\)

\(\Rightarrow B< 1-\frac{1}{2012}\Rightarrow B< 1\left(1\right)\)

\(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2012^2}\)

\(\Rightarrow B>\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+\frac{1}{4\cdot5}+...+\frac{1}{2012\cdot2013}\)

\(\Rightarrow B>\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{2012}-\frac{1}{2013}\)

\(\Rightarrow B>\frac{1}{2}-\frac{1}{2013}\Rightarrow\frac{1}{2}-\frac{1}{2013}< B\left(2\right)\)

Từ (1) và (2) => \(\frac{1}{2}-\frac{1}{2013}< B< 1\)

a)A=1+1/2²+1/3²+....+1/100²

      <1+1/1.2+1/2.3+...+1/99.100=1+1-1/2+1/2-1/3+...+1/99-1/100=2-1/100=199/200<2

b)B=1/2²+1/3²+...+1/2012²

     <1/1.2+1/2.3+...+1/2011.2012=1-1/2+1/2-1/3+...+1/2011-1/2012=1-1/2012=2011/2012

     1/2-1/2013=2011/4026<2011/2012<1

Ta có : \(A=\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+......+\frac{1}{2^{100}}\)

\(\Rightarrow4A=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^4}+.....+\frac{1}{2^{98}}\)

\(\Rightarrow4A-A=\frac{1}{2}-\frac{1}{2^{100}}\)

\(\Rightarrow3A=\frac{2^{99}-1}{2^{100}}\)

\(\Rightarrow A=\frac{2^{99}-1}{\frac{2^{200}}{3}}\)

Vì : \(\frac{2^{99}-1}{2^{200}}< 1\)

Nên : \(A< \frac{1}{3}\)

Ta thấy : \(\frac{1}{2^2}< \frac{1}{3}\)

             \(\frac{1}{2^4}< \frac{1}{3}\)

                 ...

              \(\frac{1}{2^{100}}< \frac{1}{3}\)

\(\Rightarrow A=\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}+...+\frac{1}{2^{100}}< \frac{1}{3}\)

Vậy \(A< \frac{1}{3}\)

Chúc bạn học tốt :>

A.\(4\)=\(1+\frac{1}{2^2}+\frac{1}{2^4}+...+\frac{1}{2^{98}}\)

=> 4A-A=1-\(\frac{1}{2^{100}}\)

=> A=\(\frac{1}{3}\left(1-\frac{1}{2^{100}}\right)=\frac{1}{3}-\frac{1}{3}.\frac{1}{2^{100}}< \frac{1}{3}\)

Ta có:\(4A=1+\frac{1}{2^2}+\frac{1}{2^4}+.........+\frac{1}{2^{98}}\)

\(\Rightarrow4A-A=3A=1-\frac{1}{2^{100}}< 1\Rightarrow A< \frac{1}{3}\left(ĐPCM\right)\)