Confused about the solution here – isn’t exponential growth much larger than factorial?
EDIT: Obviously we can’t use log in the GRE, in general, I mean.
ChatGPT is confirming this:
What?
Don’t use chatgpt for math lol; also, the idea of using stirling’s is overkill
Anyway, a trivial way to see this is to realize that since e^x converges then you have:
e^x = \sum_{i = 0}^{\infty} \frac{x^n}{n!} then it’s clear that n! “grows faster” than its exponential counterpart for sufficiently large n.
ChatGPT has been invaluable for me when it comes to understanding GRE questions – it does make mistakes from time to time, that’s for sure.
So are you saying that the calculation is wrong?
isn’t exponential growth much larger than factorial?
I mean that exponential growth is much quicker than a factorial? I’m not really sure which concept is tested here?
What does it mean that “since e raised to x converges”?
I mean yes it’s obviously wrong and it’s not really hard to see.
100! = \underbrace{100}_{4 \cdot 5^2} \cdot \underbrace{99}_{11 \cdot 9} \cdot \underbrace{98}_{14 \cdot 7} \ldots 4 \cdot 3 \cdot 2 \cdot 1 = 5 \cdot 9 \cdot 14 \cdot \ldots 20 \cdot 12 \cdot 22 \cdot 7
4^{100} = \underbrace{4 \cdot 4 \cdot 4 \ldots 4}_{100 \text{ times}}
Every term in 100! is now > than their counterpart in 4^{100}. The \ldots isn’t that clear in what i wrote, but it’s kinda hard to describe what i mean without more “advanced” notation, so i’ll leave at this. It should be pretty evident from context though.
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That explanation did it for me! Thank you so much!