I can simplify this equation: $$\sum_{k=0}^{20} k*\:^kP_k$$ to something like $$1*1!\;+2*2!\;+\;...\;+\;20*20!$$ But how do I simplify this equation? Please help me.
2026-04-05 14:45:00.1775400300
How to solve $\sum_{k=0}^{20} k\;\;^kP_k$
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One may observe that $$ k \cdot k!=(k+1)!-k! $$ then one may use a telescoping sum.