Factorial over Superfactorial!

\[S=\frac { 1 }{ 1! } +\frac { 1\times 2 }{ 1!\times 2! } +\frac { 1\times 2\times 3 }{ 1!\times 2!\times 3! } +\dots \]

\[{ S }_{ 0 }=\frac { 1 }{ 1! } +\frac { 1 }{ 1!\times 2! } +\frac { 1 }{ 1!\times 2!\times 3! } +\dots \]

If \[S-{ S }_{ 0 }=A{ e }^{ B }\] for integers A and B, find \[A+B\]


Write \(0.5\) if \(S\) diverges and \({ S }_{ 0 }\) converges.

Write \(1.5\) if \(S\) converges and \({ S }_{ 0 }\) diverges.

Write \(2.5\) if \(S\) and \({ S }_{ 0 }\) diverge.

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