Sum of an infinite geometric sum

Algebra Level 5

Let \(\mathcal{A}\) be the set of first hundred natural numbers.
A function \(f(t)\) is defined from \(\mathcal{A}\) to \(\mathbb{R}\) which denotes the sum of of infinite geometric series whose first term is \(\frac{t-1}{t!}\) and the common ratio is \(\frac{1}{t}\).

Let \(S(n)\) denote the sum:
\[S(n) = \sum_{r=1}^n | (r^2-3r+1)f(r) |\]

Find the value of \(\frac{100^2}{100!} + S(100)\).


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