Three Proofs of the Inequality e < (1 + (1/n))n+0.5

Let us see the motivation behind the above result. The reader can observe that the limit (2) is modestly different than the classical limit (1). Let us approximate e from these two limits using n = 1000. From the classical limit, we get e ≈ 2.71692393; which is accurate only to 3 decimal places. From the new limit (2), we get e ≈ 2.71828205; which is e accurate to 6 decimal places. Thus, the new limit appears to be a big improvement over the classical result.