Question Number 101078 by Coronavirus last updated on 30/Jun/20

find  the  following   sum        Σ_(k=1) ^n C_( n) ^( k) x^(−k) (k−1)!

Answered by abdomathmax last updated on 01/Jul/20

f(u) =Σ_(k=1) ^n  C_n ^k (k−1)! u^k    (herr u =x^(−1) ) ⇒  f(u) = Σ_(k=1) ^n  ((n!)/(k!(n−k)!))(k−1)! u^k   =n! Σ_(k=1) ^(n )  (u^k /(k(n−k)!)) ⇒f^′ (u)=n!Σ_(k=1) ^n  (u^(k−1) /((n−k)!))  =_(n−k =p)   n! Σ_0 ^(n−1)  (u^(n−p−1) /(p!))  =n! u^(n−1)  Σ_(p=0) ^(n−1)   (((u^(−1) )^p )/(p!))  we have   Σ_(p=0) ^∞  (((u^(−1) )^p )/(p!)) =e^(u^(−1)  )  =Σ_(p=0) ^(n−1 ) (((u^(−1) )^p )/(p!)) +Σ_(p=n) ^∞  (((u^(−1) )^p )/(p!))  ⇒Σ_(p=0) ^(n−1 )  (((u^(−1) )^p )/(p!)) =e^(1/(u )) −Σ_(p=n) ^(∞ )  (u^(−p) /(p!))  ⇒f^′ (u) =n! u^(n−1)  (e^(1/u)  −Σ_(p=n) ^∞  (u^(−p) /(p!))) ⇒  f(u) =n! ∫  u^(n−1)  e^(1/u)  du−n!Σ_(p=n) ^∞  (1/(p!)) ∫  u^(n−1−p)  du +c  be continued....