How can I tell which one of these numbers is greater?

Question

I have two very large numbers how do I tell which one is greater. The two numbers are $$\sum^{9(10^{99}) } _{i=1}i^9$$ and

$9^{9^{9^{9^{9^{9^{9^{9^{9^{9}}}}}}}}}$

Answer 1

The power tower is much larger. Note that you have to solve it from above.

So, you start with $9^9$ . Then you take $9$ to the power of that number and so on. If you are at $9\uparrow 9\uparrow 9\uparrow 9$, you will have left the sum in the dust because it is smaller than $(9\times 10^{99})\times (9\times 10^{99})^9=(9\times 10^{99})^{10}=9^{10}\times 10^{990}$

Answer 2

$$\sum^{9\times10^{99} } _{i=1}i^9 <\sum^{9\times10^{99} } _{i=1} \left( 9\times10^{99}\right)^9=\left( 9\times10^{99}\right)^{10}<9^{2000}$$

OP can you take it from here? Hint: $9^9>2000$