<s>[INST] $\frac{n}{2} < 1+\frac{1}{2} + \frac{1}{3}+ \ldots +\frac{1}{2^n-1}How to show that for $n\geqslant 2$
$$\frac{n}{2} < 1+\frac{1}{2} + \frac{1}{3}+ \ldots +\frac{1}{2^n-1}<n$$
 [/INST] You can group the sum as follows:
$$\sum_{k=1}^{2^n}{\frac{1}{k}}> 1 + \frac{1}{2} + \left(\frac{1}{4} + \frac{1}{4}\right) + \left(\frac{1}{8} + \frac{1}{8}+ \frac{1}{8}+ \frac{1}{8}\right)...$$
So that:
$$\sum_{k=1}^{2^n-1}{\frac{1}{k}} > 1 + \frac{n}{2}-\frac{1}{2^n}>\frac{n}{2}$$
If instead you group them by the lower power of two you easily get the other limit.
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