\(\dfrac{9}{10\text{!}}+\dfrac{9}{11\text{!}}+...+\dfrac{9}{1000\text{!}}=\dfrac{10-1}{10\text{!}}+\dfrac{10-1}{11\text{!}}+...+\dfrac{10-1}{1000\text{!}}\)

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

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

\(=\dfrac{1}{9\text{!}}-\dfrac{1}{1000\text{!}}< \dfrac{1}{9\text{!}}\)

=> \(\dfrac{9}{10\text{!}}+\dfrac{9}{11\text{!}}+...+\dfrac{9}{1000\text{!}}< \dfrac{1}{9\text{!}}\)

True or False??? 

910!+911!+...+91000!=10−110!+10−111!+...+10−11000!

=1010!−110!+1011!−111!+...+101000!−11000!

=19!−110!+110!−111!+...+1999!−11000!

=19!−11000!<19!

=> 910!+911!+...+91000!<19!

True or False???  [haha]