100-sided dice roll problem When should I stop rolling if it costs $1 for each roll and I earn only the value of the final roll shown on a 100-sided dice roll? My intent is to maximise profit and I have unlimited rolls

As a stats learner some of the answers here went far above my head, but with my intuition I came to a similar conclusion so I thought I could be worth sharing my mental process in case it might help someone or to get it commented on by someone more expert.
With every new dice roll you are paying 1\$ so you want to increase the expect utility by at least 1\$.
Now let's say you already rolled a 99, you are going to make 99\$ and a new dice roll is "good" only if you get 100. The chances are $1/100$, so the expected utility is $0.01\$$. Rolling a dice again is not worth it.
What if you already got a 98? You can make 1\$ more by rolling a 99 or 2\$ more by rolling a 100. The two options are mutually exclusive and equally likely so we can just sum up their expected utility $0.01\$ + 0.02\$ = 0.03\$$. Not worth it again.
So we just need to find the value $n$ for which the sum of expect utilities is more than 1\$, which would mean, finding the $m = 100 - n$ for which $\dfrac{m*(m+1)}{2 * 100} > 1$ which is "the sum of the increases of the expected utilities for all the values greater than ours is greater than 1\$".
That gives us $n = 86$ with an increase in the expected utility of $1.05\$$.
