Inequality about the most common divergent series We know that:
$1+\frac{1}{2}+\frac{1}{3}+....$ is a divergent series.
I have a small problem about this series.
Show that:  $1+\frac{1}{2}+\frac{1}{3}+....+\frac{1}{2^{2006}} >50$
How to prove this one?
Can you use a basic knowledge to explain for a 14 years old student?

Notice $1+\frac{1}{2}+\dots + \frac{1}{n} =\int\limits_{i=1}^{n+1} \frac{1}{\lfloor x \rfloor } \geq \int\limits_{i=1}^{n+1} \frac{1}{ x  } = \ln(n+1)$
We conclude your sum is at least $\ln(2^ {2006} +1)$ and the rest is easy.

A basic way to prove it is by noticing
$\frac{1}{2^ n} + \frac{1}{2^n + 1} + \dots + \frac{1}{2^{n+1}-1}> \frac{2^ n}{2^{n+1}} = \frac{1}{2}$ and notice we can split the sum into way more than $100$ of these sums.
