<s>[INST] Tricky sequences and series problem For a positive integer $n$, let $a_{n}=\sum\limits_{i=1}^{n}\frac{1}{2^{i}-1}$. Then are the following true:
$a_{100} > 200$ and 
$a_{200} > 100$?
Any help would be thoroughly appreciated. This is a very difficult problem  for me. :(
 [/INST] $$\begin{gather}a_n = 1 + \underbrace {\frac{1}{2} + \frac{1}{3}}_{2\text{ terms}} + \underbrace {\frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7}}_{4\text{ terms}} + \ldots + \underbrace { \frac{1}{2^{n -1}} + \frac{1}{2^{n -1}+1} + \ldots +  \frac{1}{2^n -1}}_{2^{n-1}\text{ terms}}\\
> 1 + 2\cdot\frac{1}{4} + 4\cdot\frac{1}{8} + \ldots + 2^{n-1}\cdot\frac{1}{2^{n}}
\\ =1 + \underbrace { \frac{1}{2} + \frac{1}{2} + \ldots + \frac{1}{2}}_{(n-1) \text{ terms}} = 1 +\frac{n-1}{2} = \frac{n+1}{2}.
\end{gather}
$$
Thus, $a_n > \frac{n+1}{2}.$
On the other hand,
$$a_n < 1 + 2\cdot\frac{1}{2} + 4\cdot\frac{1}{4} + \ldots + 2^{n-1}\cdot\frac{1}{2^{n-1}} = 1+(n-1 )= n,$$
so the inequality $a_{100} > 200$ cannot be true.
</s>