\(A=0,2113727891\)

\(\frac{1}{6}=0,166666667\)

\(\frac{1}{4}=0,25\)

\(\Leftrightarrow\frac{1}{6}< A< \frac{1}{4}\)

Nhận xét:

\(\frac{1}{2^2}<\frac{1}{2\times3}=\frac{1}{2}-\frac{1}{3}\)

\(\frac{1}{3^2}<\frac{1}{3\times4}=\frac{1}{3}-\frac{1}{4}\)

....

\(\frac{1}{10^2}<\frac{1}{10\times11}=\frac{1}{10}-\frac{1}{11}\)

Tính tổng ta có:

\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{10^2}<\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{10}-\frac{1}{11}=\frac{1}{2}-\frac{1}{11}=\frac{9}{22}<1\)

đặt A=1/1.2+1/2.3+...+1/9.10

B=1/2^2+1/3^2+...+1/10^2

ta có:B=1/2^2+1/3^2+...+1/10^2<A=1/1.2+1/2.3+...+1/9.10

mà A=1/1.2+1/2.3+...+1/9.10

=1-1/2+1/2-1/3+...+1/9-1/10

=1-1/10<1

=>A<B<1

=>A<1

C2:\(\frac{1}{2^2}+\frac{1}{3^2}+....+\frac{1}{10^2}\)<\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{9.10}\)

=\(1-\frac{1}{10}=\frac{10}{10}-\frac{1}{10}=\frac{9}{10}<1\)

=\(\frac{9}{10}<1=\frac{9}{10}<\frac{10}{10}\)

Vậy \(\frac{9}{10}<1\)

đặt \(A=\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}\)

Ta có :

\(A< \frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{99.100}\)

\(=\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}=\frac{1}{4}-\frac{1}{100}< \frac{1}{4}\)

Lại có :

\(A>\frac{1}{5.6}+\frac{1}{6.7}+...+\frac{1}{100.101}\)

\(=\frac{1}{5}-\frac{1}{6}+\frac{1}{6}-\frac{1}{7}+...+\frac{1}{100}-\frac{1}{101}=\frac{1}{5}-\frac{1}{101}>\frac{1}{6}\)

Tu lam di

Ta có:

\(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{99.100}\)

                                                                        \(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+\frac{1}{5}-\frac{1}{6}+...+\frac{1}{99}-\frac{1}{100}\)

                                                                        \(=\frac{1}{2}-\frac{1}{100}\)

                                                                       \(=\frac{49}{100}\)

Mà \(\frac{49}{100}< \frac{1}{2}\)

Vậy \(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2}\)

Ta có:\(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{99.100}\)(1)

Xét\(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+\frac{1}{5.6}+...+\frac{1}{99.100}\)

\(=\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{99}-\frac{1}{100}\)

\(=\frac{1}{2}-\frac{1}{100}\)

\(=\frac{50}{100}-\frac{1}{100}\)

\(=\frac{49}{100}\)(2)

Mà\(\frac{49}{100}< \frac{50}{100}=\frac{1}{2}\)(3)

Từ (1), (2), (3)\(\Rightarrow\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+\frac{1}{6^2}+...+\frac{1}{100^2}< \frac{1}{2}\left(đpcm\right)\)

Vậy...

Linz

Ta có : \(\frac{1}{3^2}< \frac{1}{2.3}\)

            \(\frac{1}{4^2}< \frac{1}{3.4}\)

            \(\frac{1}{5^2}< \frac{1}{4.5}\)

             ...

            \(\frac{1}{100^2}< \frac{1}{99.100}\)

\(\Rightarrow\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{100^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\)

\(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)

\(\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{100^2}< \frac{1}{2}-\frac{1}{100}< \frac{1}{2}\)

\(\Rightarrow\frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+...+\frac{1}{100^2}< \frac{1}{2}\left(đpcm\right)\)

Ta có: 1/2^2 < 1/1.2

          1/3^2 < 1/2.3 

        .........................

.......................................

          1/100^2 < 1/99.100

Ta có: 1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 1/1.2 + 1/2.3 + 1/3.4 + ...... + 1/99.100

          1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 1 - 1/100

          1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 99/100 < 3/4

         1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 3/4

Ta có: 1/2^2 < 1/1.2

          1/3^2 < 1/2.3 

        .........................

.......................................

          1/100^2 < 1/99.100

Ta có: 1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 1/1.2 + 1/2.3 + 1/3.4 + ...... + 1/99.100

          1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 1 - 1/100

          1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 99/100 < 3/4

         1/2^2 + 1/3^2 + 1/4^2 +......+1/100^2 < 3/4

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

\(A=\frac{1}{2^2}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)\)

Đặt \(B=1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)

Ta có: 1  = 1 ; \(\frac{1}{2^2}< \frac{1}{1.2}\); \(\frac{1}{3^2}< \frac{1}{2.3}\); .... ; \(\frac{1}{50^2}< \frac{1}{49.50}\)

\(\Rightarrow B< 1+\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{49.50}=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)

\(\Rightarrow B< 2-\frac{1}{50}< 2\)\(\Rightarrow B< 2\)

\(\Rightarrow A=\frac{1}{2^2}.B< \frac{1}{2^2}.2=\frac{1}{2}\)

\(\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{100^2}\)

\(=\frac{1}{2^2}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)\)

Đặt \(A=1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\)

\(< 1+\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

\(=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)

\(=2-\frac{1}{50}< 2\)

\(\Rightarrow\frac{1}{2^2}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)< \frac{1}{2^2}.2=\frac{1}{2}\)