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Question Number 138600 by Raxreedoroid last updated on 15/Apr/21

prove or disprove  Σ_(k=1) ^n f(k)=f(1)+Σ_(k=2) ^n (((Σ_(i=1) ^(k−1) (−1)^(i+1) f(i+1)C_(i−1) ^(k−2) )/((k−1)!)) Π_(i=1) ^(k−1) (n−i))

$$\mathrm{prove}\:\mathrm{or}\:\mathrm{disprove} \\ $$$$\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}{f}\left({k}\right)={f}\left(\mathrm{1}\right)+\underset{{k}=\mathrm{2}} {\overset{{n}} {\sum}}\left(\frac{\underset{{i}=\mathrm{1}} {\overset{{k}−\mathrm{1}} {\sum}}\left(−\mathrm{1}\right)^{{i}+\mathrm{1}} {f}\left({i}+\mathrm{1}\right){C}_{{i}−\mathrm{1}} ^{{k}−\mathrm{2}} }{\left({k}−\mathrm{1}\right)!}\:\underset{{i}=\mathrm{1}} {\overset{{k}−\mathrm{1}} {\prod}}\left({n}−{i}\right)\right) \\ $$

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