can you also give me the link of the paper that proved for b>2000 we will have :
3^b-2.99^b > |2^a-3^b| > 2.99^b ?
It depends on the \epsilon (or \epsilon_n) you use (if you use the one from my post, you’ll end-up with the right side converging to 2.99), but if you start with 3^n>\lambda_n>\frac{3^n}{n^{13.3}} you can say that the n^{th} root limit is 3 (also for 3^n-\lambda_n)
I have no paper explaining that, that’s based on the math exchange post linked:
Just replace the 2^n part with 2.99^ n so that
\frac{1}{n^a}>\frac{1}{(\frac{3}{2.99})^n} which is true for n>48468 (instead of 196) and a manual check showing it’s true for n>2000 (instead of 5)
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