Lack of a response tells me no. Can anyone give me advice on approximating n?
It would be hard to find a necessary and sufficient condition on n for this inequality to hold. However, problems like this usually occur in a first course on analysis, and the aim there is not to find the smallest value of n that will satisfy the inequality, but to find some number N such that all values of n gretaer than N will satisfy the inequality. In that case, you can make the problem much easier by getting a rough upper bound for the left side of the inequality.
For example, . If you can find a value of N such that whenever n>N, it will follow that the original inequality will hold whenever n>N. (Here, K stands for some given large number, such as 1000001.)
But , so take N to be any integer greater than and your inequality will hold for all n>N.
Just to emphasise, is not the smallest number with that property. If you want to find that number, you can use a computer to check all the values of N from 1 to , and it will tell you which is the first one to make the inequality true.