Which is the bigger order of growth ? (Big-O)

Question

nⁿ or 2ⁿ². I think that the n term will grow faster than the 2 term and the n² will grow faster than the n term, but overall will the n² exponent cause the second to grow faster.

Answer 1

Let's get ln from both sides:

log (n ^ n) = n log n
log (2 ^ (n ^ 2)) = n ^ 2

Obviously n ^ 2 growths faster than n log n. It means that n ^ n growths faster than 2 ^ (n ^ 2).