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Euler's use of infinitesimal calculus

Can anyone explain (12) in this article. Why can we write $f(n)$ as a sum of derivatives: $$f(n) = \frac{ds}{dn} - \frac{1}{2!}\frac{d^2s}{dn^2} + \frac{1}{3!}\frac{d^3s}{dn^3} - \cdots$$ where $s(n) ...