[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Prove that the harmonic series [math]\sum_{k=1}^\infty \frac1k[/math] diverges using the following elementary argument. Begin by grouping the terms of the series:

[[math]] \sum_{k=1}^\infty \frac1k = 1 + \left(\frac12+\frac13\right) + \left(\frac14+\frac15+\frac16+\frac17\right) [[/math]]

[[math]] + \left(\frac18+\frac19+\frac1{10}+\frac1{11}+\frac1{12}+ \frac1{12}+\frac1{13}+\frac1{14}+\frac1{15}\right) + \cdots , [[/math]]

and observe that

[[math]] \frac12 + \frac13 \gt \frac14 + \frac14 = \frac12 , [[/math]]

[[math]] \frac14+\frac15+\frac16+\frac17 \gt \frac18+\frac18+\frac18+\frac18=\frac12, \quad \mbox{etc.} [[/math]]