Calculate LIM (n →∞) ∑ upper n lower k = 1 (K + 2) / [K! + (K + 1)! + (K + 2)!]

Calculate LIM (n →∞) ∑ upper n lower k = 1 (K + 2) / [K! + (K + 1)! + (K + 2)!]

The original formula = ∑ (K + 2) / [K! (1 + K + 1 + (K + 1) (K + 2)] = ∑ 1 / (k! (K + 2))
Let s (x) = ∑ 1 / K! (K + 2) * x ^ (K + 2), obviously s (0) = 0
S'(x)=∑1/k!x^(k+1)=x∑1/k!*x^k
=x(e^x-1)
S(X)=∫x(e^x-1)dx=∫xe^xdx-∫xdx
=xe^x-e^x-x^2/2+c
S(0)=-1+C=0
∴C=1
S(X)=xe^x-e^x-x^2/2+1
Let x = 1
Original stage = e-e-1 / 2 + 1 = 1 / 2