The positive integer just greater than S=(1+0.0001)10000 is

Let n=10000, thenS=1+1nn=1+nC11n+nC21n2+⋯+nCn1nn=1+1+12!n(n−1)n2+13!n(n−1)(n−2)n3+⋯+1n!n(n−1)⋯(n−n+1)nn =1+1+12!1−1n+13!1−1n1−2n+⋯+1n!1−1n⋯1−n−1n          <1+1+12!+13!+⋯+1n!But 1r!=11.2…r≤12r−1 ∀r≥2.Thus, S≤1+1+12+122+⋯+12n−1           =1+1−(1/2)n1−1/2=3−12n−1<3.