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Question Number 30194 by abdo imad last updated on 17/Feb/18

study the convergence of u_n = Σ_(k=0) ^n (1/C_n ^k ) with C_n ^k =((n!)/(k!(n−k)!)) .

studytheconvergenceofun=∑k=0n1CnkwithCnk=n!k!(n−k)!.

Commented by prof Abdo imad last updated on 22/Feb/18

let put x_n = (1/C_n ^k ) ∀ n∈ N x_n >0 we have (x_(n+1) /x_n ) = (C_n ^k /C_(n+1) ^k ) = (((n!)/(k!(n−k)!))/(((n+1)!)/(k!(n+1−k)!)))= (1/(n+1)) (((n+1−k)!)/((n−k)!)) = ((n+1−k)/(n+1))= 1− (k/(n+1)) <1 from that u_n isconvergent.

letputxn=1Cnk∀n∈Nxn>0wehavexn+1xn=CnkCn+1k=n!k!(n−k)!(n+1)!k!(n+1−k)!=1n+1(n+1−k)!(n−k)!=n+1−kn+1=1−kn+1<1fromthatunisconvergent.

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