Do factorials really grow faster than exponential functions? [closed]

Having trouble understanding this. Is there anyway to prove it?

Answer

If you’re not quite in the market for a full proof:

an=a×a×a×a...×a
n!=1×2×3×4…×n

Now what happens as n gets much bigger than a? In this case, when n is huge, a will have been near some number pretty early in the factorial sequence. The exponential sequence is still being multiplied by that (relatively tiny) number at each step, while n! is being multiplied by n. So even if n! starts out small, it’ll eventually start being multiplied by gigantic numbers at each step, and quickly outgrow the exponential. If a=10 and n=100, then an has around 100 digits, while n! has over 150 digits. Note that near n=100, n! is having roughly 2 digits added per step (and that rate will only increase), while an is still only ever going to get one more with every step. No contest.

Attribution
Source : Link , Question Author : Billy Thompson , Answer Author : Robert Mastragostino

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