The coefficient of  xnin the expansion of1+11!x+12!x2…+1n!xn2

Coefficient of xn in1+11!x+12!x2+…+1n!xn2= coefficient of  xn is1+11!x+12!x2+…+1n!xn+1(n+1)!xn+1…2= coefficient of xn in  =   e2x=1+2x+22x22!+23x33!⋯=2nn!Alternate Solution Coefficient of  xn in1+11!x+12!x2+⋯+1n!xn 1+11!x+12!x2+⋯+1n!xn =1n!+11!(n−1)!+12!(n−2)!+⋯+ =1n! nC0+nC1+nC2+⋯nCn−1+nCn=2nn!