Category Archives: Analysis

Recall that the harmonic series ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{1}{n}}$ diverges. This is because we may bound the partial sums below, like so: $$1+\frac{1}{2}+(\frac{1}{3}+\frac{1}{4})+(\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8})+\cdots >$$ $$\frac{1}{2}+\frac{1}{2}+(\frac{1}{4}+\frac{1}{4})+(\frac{1}{8}+\frac{1}{8}+\frac{1}{8}+\frac{1}{8})+\cdots =$$ $$\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\cdots \to \mathrm{\infty }$$ We may replace $n$ by $an+b$, where $a,b\in \mathbb{R}$, $a$ and $b$ not both $0$, to retain a … Continue reading →