Series Math Example 2

Solution

1. 1
   Group terms: $1 + \frac{1}{2} + \left(\frac{1}{3}+\frac{1}{4}\right) + \left(\frac{1}{5}+\cdots+\frac{1}{8}\right) + \cdots$
2. 2
   Each group of $2^k$ terms satisfies: $\sum_{n=2^k+1}^{2^{k+1}} \frac{1}{n} > 2^k \cdot \frac{1}{2^{k+1}} = \frac{1}{2}$.
3. 3
   Since infinitely many groups each exceed $\frac{1}{2}$, the partial sums grow without bound.
4. 4
   Therefore the series diverges.