EXAMPLE 2 Prove that 6ex is equal to the sum of its Maclaurin series. SOLUTION If f(x)=6ex, then f(n+1)(x)= for all n. If d is any positive number and xd, then f(n+1)(x)=6ed. So Taylor's Inequality, with the series centered at a =0 and with upper bound M=6e, says that Notice that the same constant M=6ed works for every value of n. But, from this equation, we have limn(n+1)!6edxn+1=6edlimn(n+1)!xn+1= It follows from the Squeeze Theorem that limnRn(x)=0 and therefore limnRn(x)= for all values of x. By this theorem, 6ex is equal to the sum of its Maclaurin series, that is, 6ex=n=0mn!6xnforallx