# How to find the minimum of n for which ((2^n)-1)lna/(2^(n-1)*n) < 0.01*lna ?

Dec 6, 2017

$n \ge 200$

#### Explanation:

$\frac{\left({2}^{n} - 1\right) \ln a}{{2}^{n - 1} \cdot n} < 0.01 \cdot \ln a$

Assuming $\ln a > 1$ we have

${2}^{n} - 1 < \frac{0.01}{2} {2}^{n} n$ or

${2}^{n} \left(1 - \frac{n}{200}\right) < 1 \Rightarrow n \ge 200$