Approximation formulas for the constant e and an improvement to a Carleman-type inequality

We give an explicit formula for the determination of the coefficients cj appearing in the expansionx(1+∑j=1qcjxj)(πΓ(x+12))1/x=e+O(1xq+1) for x→∞ and q∈N:={1,2,…}. We also derive a pair of recurrence relations for the determination of the constants λℓ and μℓ in the expansion(1+1x)x∼e(1+∑ℓ=1∞λℓ(x+μℓ)2ℓ−1) as x→∞. Based on this expansion, we establish an inequality for (1+1/x)x. As an application, we give an improvement to a Carleman-type inequality.