Reckon you can solve this recursion?

Calculus Level 5

Consider a sequence \((a_n)_{n=1}^\infty \) with \(a_1 = 1, a_2 = 2\). Let \(\displaystyle S_n = \sum_{j=1}^n a_n \) such that it satisfy the recurrence relation \(a_n = n\times S_{n-1} \) for \(n=3,4,5,\ldots \).

Lastly, denote \(s\) as the value of \( \displaystyle \sum_{n=1}^\infty \dfrac n{a_n} \).

Find the last three digits of \(\left\lfloor 10^8 \times s \right\rfloor \).

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