# I was asked to find an explicit formula for S n </msub> = <munderover> &#x221

I was asked to find an explicit formula for
${S}_{n}=\sum _{r=1}^{n}\frac{1}{r\left(r+1\right)}$
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Oswaldo Rosales
${S}_{n}=\sum _{r=1}^{n}\frac{1}{r\left(r+1\right)}\phantom{\rule{0ex}{0ex}}=\sum _{1}^{n}\left(\frac{1}{r}-\frac{1}{r+1}\right)\phantom{\rule{0ex}{0ex}}=\left(\frac{1}{1}-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\cdots +\left(\frac{1}{n-1}-\frac{1}{n}\right)+\left(\frac{1}{n}-\frac{1}{n+1}\right)\phantom{\rule{0ex}{0ex}}=1-\frac{1}{n+1}$