Giải thích các bước giải:

Đặt A= 1/4+1/16+1/36+1/64+1/100+1/144+1/196

= 1/2^2+ 1/4^2+ 1/6^2+….+ 1/16^2

= 1/2^2.( 1+ 1/2^2+ 1/3^2+…+ 1/8^2)

Ta có 1+ 1/2^2+ 1/3^2+…+ 1/8^2< 1+ 1/1.2+ 1/2.3+….7.8= 1+ 1-1/2+ 1/2- 1/3+….+ 1/7- 1/8

= 2- 1/8< 2

Nên ( 1+ 1/2^2+ 1/3^2+…+ 1/8^2)< 2

=> A< 1/2^2 nhân 2= 1/2

Vậy A< 1/2

Đặt \(A=\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+...+\dfrac{1}{196}\)

\(=\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{13^2}\)

Đặt \(B=\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{12\cdot13}\)

Ta có:

\(\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{13^2}\)\(<\)\(B=\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{12\cdot13}\left(1\right)\)

Mà \(B=\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{12\cdot13}\)

\(=\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{12}-\dfrac{1}{13}\)

\(=\dfrac{1}{2}-\dfrac{1}{13}< \dfrac{1}{2}\left(2\right)\). Từ \((1)\) và \((2)\) ta có:

\(A< B< \dfrac{1}{2}\Rightarrow A< \dfrac{1}{2}\) (Điều phải chứng minh)

bài này có trong sách Nâng cao và Phát triển bạn nhé

Đặt \(A=\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+\dfrac{1}{64}+...+\dfrac{1}{10000}\)

Ta có:

\(A=\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+\dfrac{1}{64}+...+\dfrac{1}{10000}\)

\(\Rightarrow A=\dfrac{1}{4}\left(1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{50^2}\right)\)

\(\Rightarrow A< \dfrac{1}{4}\left(1+\dfrac{1}{1.2}+\dfrac{1}{2.3}+...+\dfrac{1}{49.50}\right)\)

\(\Rightarrow A< \dfrac{1}{4}\left(1+1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{49}-\dfrac{1}{50}\right)\)

\(\Rightarrow A< \dfrac{1}{4}\left(1+1-\dfrac{1}{50}\right)\)

\(\Rightarrow A< \dfrac{1}{4}.\dfrac{99}{50}\)

\(\Rightarrow A< \dfrac{99}{200}< \dfrac{1}{2}\)

Vậy \(\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+\dfrac{1}{64}+...+\dfrac{1}{10000}< \dfrac{1}{2}\) (Đpcm)

\(\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+...+\dfrac{1}{10000}=\dfrac{1}{2^2}+\dfrac{1}{4^2}+\dfrac{1}{6^2}+...+\dfrac{1}{100^2}\)

\(=\dfrac{1}{2^2}\left(1+\dfrac{1}{2^2}+\dfrac{1}{3^2}+...+\dfrac{1}{50^2}\right)< \dfrac{1}{4}\left(1+\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}+...+\dfrac{1}{49.50}\right)=\dfrac{1}{4}\left(1+1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{49}-\dfrac{1}{50}\right)=\dfrac{1}{4}\left(1+1-\dfrac{1}{50}\right)=\dfrac{1}{4}\left(2-\dfrac{1}{50}\right)< \dfrac{1}{2}\)

\(\Rightarrow\dfrac{1}{4}+\dfrac{1}{16}+\dfrac{1}{36}+...+\dfrac{1}{10000}< \dfrac{1}{2}\)

Cho A = 1/2 .3/4.5/6.....199/200.Chứng tỏ rằng B mũ 2 <1/201.Bạn có làm dược ko ?

bài 2

a) =

b) >

c) =

d) =

a) =

b) >

c) =

d) <

a) =

b) >

c) =

d) =

11,6

= 1,1 + 1,2 + 1,3 + 1,4 + 1,5 + 1,6 + 1,7 + 1,8 + 1,9
= 2,3 + 1,3 + 1,4 + 1,5 + 1,6 + 1,7 + 1,8 + 1,9
= 3,6 + 1,4 + 1,5 + 1,6 + 1,7 + 1,8 + 1,9
= 5 + 1,5 + 1,6 + 1,7 + 1,8 + 1,9
= 6,5 + 1,6 + 1,7 + 1,8 + 1,9
= 8,1 + 1,7 + 1,8 + 1,9
= 9,8 + 1,8 + 1,9
= 11,6 + 1,9
= 13,5