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$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \cdots + \frac{99}{100!} = 1 - \frac{1}{A!}$

What is $A$?

$$ Notation: $!$ is the factorial notation. For example, $8! = 1\times2\times3\times\cdots\times8$.

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