\documentclass{jarticle}
\usepackage{amsmath,emath,emathP,epic,eepic}

\begin{document}

\begin{align*}
& \sum_{k=1}^n  [1/k - 1/(k+1)] \\
&= [1-1/2]+[1/2-1/3]+\cdots+[1/n-1/(n+1)] \\
&= [1-\teisei{1/2}]+[\teisei{1/2}-\teisei{1/3}]+[\teisei{1/3}-\teisei{1/4}]+\cdots+[\teisei{1/n}-1/(n+1)]  \\
&= [1-\teisei{\frac 12}]+[\teisei{\frac 12}-\teisei{\frac 13}]+[\teisei{\frac 13}-\teisei{\frac 14}]+\cdots+[\teisei{\frac 1n}-1/(n+1)]  \\
&=1-1/(n+1)
\end{align*}

\end{document}