[Math] Which is greater: $1000^{1000}$ or $1001^{999}$

Question

Question:

Find the greater number: $1000^{1000}$ or $1001^{999}$

My Attempt:

I know that: $(a+b)^n \geq a^n + a^{n-1}bn$.

Thus, $(1+999)^{1000} \geq 999001$

And $(1+1000)^{999} \geq 999001$

But that doesn't make much sense.

I want some hints regarding how to solve this problem.

Thanks.

Answer 1

Look at the quotient $$ \frac{1001^{999}}{1000^{1000}}=\frac1{1001}\underbrace{\left(1+\frac1{1000}\right)^{1000}}_{\approx e}$$