bài này dài lắm đó bạn mk pít lm phần a nhưng k có thời gian

Lời giải:

Ta có:
$\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+...+\frac{1}{20}< \frac{1}{11}+\frac{1}{11}+\frac{1}{11}+...+\frac{1}{11}=\frac{10}{11}<1$

Ta có điều phải chứng minh

\(1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{19}-\dfrac{1}{20}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{3}-\dfrac{1}{4}+...+\dfrac{1}{19}-\dfrac{1}{20}+\left(\dfrac{1}{2}-\dfrac{1}{2}\right)+\left(\dfrac{1}{4}-\dfrac{1}{4}\right)+...+\left(\dfrac{1}{20}-\dfrac{1}{20}\right)\)

\(=1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{20}-2\left(\dfrac{1}{2}+\dfrac{1}{4}+...+\dfrac{1}{20}\right)\)

\(=1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{20}-\left(1+\dfrac{1}{2}+\dfrac{1}{3}+...+\dfrac{1}{10}\right)\)

\(=\dfrac{1}{11}+\dfrac{1}{12}+...+\dfrac{1}{20}\) (đpcm)

Ta có: \(\dfrac{3}{10}>\dfrac{3}{15}\)

\(\dfrac{3}{11}>\dfrac{3}{15}\)

\(\dfrac{3}{12}>\dfrac{3}{15}\)

\(\dfrac{3}{13}>\dfrac{3}{15}\)

\(\dfrac{3}{14}>\dfrac{3}{15}\)

Do đó: \(\dfrac{3}{10}+\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}>\dfrac{3}{15}+\dfrac{3}{15}+\dfrac{3}{15}+\dfrac{3}{15}+\dfrac{3}{15}=1\)

hay 1<S(1)

Ta có: \(\dfrac{3}{11}< \dfrac{3}{10}\)

\(\dfrac{3}{12}< \dfrac{3}{10}\)

\(\dfrac{3}{13}< \dfrac{3}{10}\)

\(\dfrac{3}{14}< \dfrac{3}{10}\)

Do đó: \(\dfrac{3}{11}+\dfrac{3}{12}+\dfrac{3}{13}+\dfrac{3}{14}< \dfrac{3}{10}+\dfrac{3}{10}+\dfrac{3}{10}+\dfrac{3}{10}=\dfrac{12}{10}\)

\(\Leftrightarrow S< \dfrac{15}{10}=\dfrac{3}{2}< 2\)(2)

Từ (1) và (2) suy ra 1<S<2(đpcm)

Bạn làm hay quá

\(\dfrac{5}{2x1}+\dfrac{4}{1x11}+\dfrac{3}{11x2}+\dfrac{1}{2x15}+\dfrac{13}{15x4}+\dfrac{15}{4x13}\)

=7x(\(\dfrac{5}{2x7}+\dfrac{4}{7x11}+\dfrac{3}{11x14}+\dfrac{1}{14x15}+\dfrac{13}{15x28}+\dfrac{15}{28x43}\))

=7x\(\dfrac{1}{2}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{14}+\dfrac{1}{14}-\dfrac{1}{15}+\dfrac{1}{15}-\dfrac{1}{28}+\dfrac{1}{28}-\dfrac{1}{43}\)=7x(\(\dfrac{1}{2}-\dfrac{1}{43}\))

=7x\(\dfrac{41}{86}\)

=\(\dfrac{287}{86}\)

5/2x1+4/1x11+3/11x2+1/2x15+13/15x4+15/4x43=7x(5/2x7+4/7x11+3/11x14+1/14x15+13/15x28+15/28x43)=7x(1/2-1/7+1/7-1/11+1/11-1/14+1/14+1/15+1/15-1/28+1/28-1/43)=7x(1/2-1/43)=7x41/86=287/86

Xin lỗi nha trễ tí

Trang 2 nè nhìn lại cho rõ nha

giúp mk với mọi người lm ơn đó

a, Ta có :

\(M=\dfrac{1}{1\cdot2}+\dfrac{1}{1\cdot2\cdot3}+\dfrac{1}{1\cdot2\cdot3\cdot4}+...+\dfrac{1}{1\cdot2\cdot3\cdot...\cdot100}\\ < \dfrac{1}{1\cdot2}+\dfrac{1}{2\cdot3}+\dfrac{1}{3\cdot4}+...+\dfrac{1}{99\cdot100}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-...+\dfrac{1}{99}-\dfrac{1}{100}\\ =1-\dfrac{1}{100}=\dfrac{99}{100}< 1\\ \Rightarrow M< 1\\ \RightarrowĐpcm\)