Question

[tran van danh]

CMR: \(\frac{9}{10!}+\frac{9}{11!}+\frac{9}{12!}+...+\frac{9}{1000!}< \frac{1}{9!}\)

Answer 1

[Nguyễn Quỳnh Chi]

Có: \(\frac{9}{10!}=\frac{9}{10!}\)

\(\frac{9}{11!}< \frac{10}{11!}=\frac{11-1}{11!}=\frac{11}{11!}-\frac{1}{11!}=\frac{1}{10!}-\frac{1}{11!}\)

\(\frac{9}{12!}< \frac{11}{12!}=\frac{12-1}{12!}=\frac{12}{12!}-\frac{1}{12!}=\frac{1}{11!}-\frac{1}{12!}\)

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

\(\frac{9}{1000!}< \frac{999}{1000!}=\frac{1000-1}{1000!}=\frac{1000}{1000!}-\frac{1}{1000!}=\frac{1}{999!}-\frac{1}{1000!}\)

\(\Rightarrow\frac{9}{10!}+\frac{9}{11!}+\frac{9}{12!}+...+\frac{1}{1000!}< \frac{9}{10!}+\frac{1}{10!}-\frac{1}{11!}+\frac{1}{11!}-\frac{1}{12!}+...+\frac{1}{999!}-\frac{1}{1000!}\)

\(\Rightarrow\frac{9}{10!}+\frac{9}{11!}+...+\frac{1}{1000!}< \frac{10}{10!}-\frac{1}{1000!}=\frac{1}{9!}-\frac{1}{1000!}< \frac{1}{9!}\)

\(\Rightarrow\frac{9}{10!}+\frac{9}{11!}+...+\frac{9}{1000!}< \frac{1}{9!}\)

\(\Rightarrowđpcm\)

Answer 2

[đặng tuấn đức]

đặt tên là B

$\mathrm{B\; =}\frac{9}{10!}+\frac{9}{11!}+\frac{9}{12!}+.............+\frac{9}{1000!}$

Ta thấy :

$\frac{9}{10!}=\frac{10-1}{10!}=\frac{1}{9!}-\frac{1}{10!}$

$\frac{9}{11!}<\frac{11-1}{11!}=\frac{1}{10!}-\frac{1}{11!}$

$\frac{9}{1000!}<\frac{1000-1}{1000!}=\frac{1}{999!}-\frac{1}{1000!}$

$\Rightarrow B<\frac{1}{9!}-\frac{1}{10!}+\frac{1}{10!}-\frac{1}{11!}+........+\frac{1}{999!}-\frac{1}{1000!}$

$B<\frac{1}{9!}-\frac{1}{1000!}$

$\Rightarrow B<\frac{1}{9!}\to đpcm$