Question

CMR:

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

Answer 1

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

\(=\frac{10-1}{10!}+\frac{11-2}{11!}+...+\frac{1000-991}{1000!}\)

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

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

\(=\frac{1}{9!}-\frac{1}{1000!}< \frac{1}{9!}\left(đpcm\right)\)