# Sum of an infinite geometric sum

Algebra Level 4

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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