Harmonic Series

The harmonic series is the series $$\sum_{n=1}^{\infty} \frac{1}{n} = 1 +\frac{1}{2} +\frac{1}{3}+\cdots$$. The terms get smaller and smaller, and if you add 250,000,000 terms the sum is still less than 20. Therefore, you might be surprised to find out that the sequence of partial sums is unbounded and the series diverges. It was proved to diverge around 1350 by Oresme, forgotten and rediscovered again in the late $17^{th}$ century. An interesting question is how much of the sequence must be removed before the series converges?

1. Show that $1 +\frac{1}{2} +\frac{1}{3}+ \cdots+ \frac{1}{n}$ can be made larger than any real number by choosing an appropriate $n$.
2. Show that when all the terms with denominators that are not prime are removed, the series still diverges. In other words the sum of the reciprocals of the prime numbers diverges. For proofs see [5] [6] or [14].
3. Show that when all the terms that contain the digit nine are deleted, the series converges. In fact, the sum is less than 90. See An Intriguing Series in [7].
4. $2002 = 2\cdot7\cdot11\cdot13$. Find the sum of all the unit fractions that have denominators with only factors from the set $\{2,7,11,13\}$. That is, find the following sum:
   $\frac{1}{2}+\frac{1}{4}+\frac{1}{7}+\frac{1}{8}+\frac{1}{11}+\frac{1}{13}+\ frac{1}{14}+\frac{1}{16}+ \frac{1}{22}+\frac{1}{26}+\frac{1}{28}+\cdots$