Given that n n and tn nn which grows faster Prove it You m

Given that (n) = n! and t(n) = n^n, which grows faster? Prove it. You may not use Stirling\'s Approximation.

Solution

You can use logrithimic approach in order to predict which one having faster order of growth

a^n<n!

for large values of n use logrithimic approch.

Take the log of both sides, you get

nlog(a)=log(a^n)<log(n!)=i=1 to nlog(i)

Now in order to increment n you must have to only add log(a) term to the left side, but the term added to right side log(n+1) is basically large as the value n becomes large. And as the result the value becomes large to the level to make easy differentiate and be greater than nlog(a). So, Factorial is faster than Exponential