Ta có:

\(\frac{3}{5.2!}+\frac{3}{5.3!}+\frac{3}{5.4!}+...+\frac{3}{5.100!}\)

\(=\frac{3}{5}.\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)\)

\(< \frac{3}{5}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{99.100}\right)\)

\(=\frac{3}{5}.\left(1-\frac{1}{100}\right)\)

\(< \frac{3}{5}.1=\frac{3}{5}=0,6\)

bang nhau

Ta có:

3/5.2!+3/5.3!+.......+3/5.100!

=3/5(1/2!+1/3!+.......+1/100!)

< 3/5(1/1.2+1/2.3+........+1/99.100)

=3/5.(1-1/100)

<3/5=0.6

=> tổng trên<0,6

Theo đầu bài ta có:
\(\frac{3}{5\cdot2!}+\frac{3}{5\cdot3!}+\frac{3}{5\cdot4!}+...+\frac{3}{5.100!}< 0,6\)
\(\Rightarrow\frac{3}{5}\cdot\frac{1}{2!}+\frac{3}{5}\cdot\frac{1}{3!}+\frac{3}{5}\cdot\frac{1}{4!}+...+\frac{3}{5}\cdot\frac{1}{100!}< \frac{3}{5}\)
\(\Rightarrow\frac{3}{5}\cdot\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}\right)< \frac{3}{5}\)
\(\Rightarrow\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}< 1\)( điều cần chứng minh )
Mà \(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+\frac{1}{3\cdot4}+...+\frac{1}{99\cdot100}\)
\(\Rightarrow\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{99}-\frac{1}{100}\)
\(\Rightarrow\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+...+\frac{1}{100!}< 1-\frac{1}{100}< 1\)( đã chứng minh được )
Vậy \(\frac{3}{5\cdot2!}+\frac{3}{5\cdot3!}+\frac{3}{5\cdot4!}+...+\frac{3}{5\cdot100!}< 0,6\)( đpcm )

có A= \(\frac{3}{5.2!}\)+\(\frac{3}{5.3!}\)+...+\(\frac{3}{5.100!}\)=\(\frac{3}{5}\)(\(\frac{1}{2!}\)+\(\frac{1}{3!}\)+....+\(\frac{1}{100!}\))

đặt vế trong ngoặc là B. Đặt \(\frac{1}{2!}\)+\(\frac{2}{3!}\)+...+\(\frac{99}{100!}\)=C ta có C=\(\frac{2-1}{2!}\)+\(\frac{3-1}{3!}\)+....+\(\frac{100-1}{100!}\)

=\(\frac{2}{2!}\)-\(\frac{1}{2!}\)+\(\frac{1}{2!}\)-\(\frac{1}{3!}\)+...+\(\frac{1}{99!}\)-\(\frac{1}{100!}\)=1-\(\frac{1}{100!}\)<1

mà \(\frac{1}{2!}\)=\(\frac{1}{2!}\);\(\frac{1}{3!}\)<\(\frac{2}{3!}\);....;\(\frac{1}{100!}\)<\(\frac{99}{100!}\)\(\Rightarrow\)B<C<1\(\Rightarrow\)B.\(\frac{3}{5}\)<1.\(\frac{3}{5}\)=\(\frac{3}{5}\)=0.6\(\Rightarrow\)A<0.6

Cũng đơn giản mà em nhớ k cho chị nha !

vế trái được viết dưới dạng :

 \(\frac{3}{5}.\left(\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+....+\frac{1}{100!}\right)< \frac{3}{5}.\left(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{99.100}\right)\)

\(=\frac{3}{5}\left(1-\frac{1}{100}\right)< \frac{3}{5}=0,6\)