All Harmonic Series Diverge — And a Consequence!

Recall that the harmonic series $\displaystyle{\sum_{n=1}^{\infty}\frac{1}{n}}$ diverges. This is because we may bound the partial sums below, like so: $1+\frac{1}{2}+\left(\frac{1}{3}+\frac{1}{4}\right)+\left(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}\right)+\cdots >$ $\frac{1}{2}+\frac{1}{2}+\left(\frac{1}{4}+\frac{1}{4}\right)+\left(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8}\right)+\cdots=$ $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots\to\infty$ We may replace $n$ by $an+b$, whe