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....