summation of difference
- Thread starter carlacvds
- Start date
pka
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- Jan 29, 2005
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\(\displaystyle \L\begin{array}{rcl}
S_4 & = & \sum\limits_{k = 1}^4 {\left( {\frac{1}{k} - \frac{1}{{k + 2}}} \right)} \\
& = & \left( {1 - \frac{1}{3}} \right) + \left( {\frac{1}{2} - \frac{1}{4}} \right) + \left( {\frac{1}{3} - \frac{1}{5}} \right) + \left( {\frac{1}{4} - \frac{1}{6}} \right) \\
& = & \left( {1 + \frac{1}{2}} \right) - \left( {\frac{1}{5} + \frac{1}{6}} \right) \\
\end{array}\)
\(\displaystyle \L\begin{array}{rcl}
S_n & = & \sum\limits_{k = 1}^n {\left( {\frac{1}{k} - \frac{1}{{k + 2}}} \right)} \\
& = & \left( {1 - \frac{1}{3}} \right) + \left( {\frac{1}{2} - \frac{1}{4}} \right) + \left( {\frac{1}{3} - \frac{1}{5}} \right) + \cdots + \left( {\frac{1}{n} - \frac{1}{{n + 2}}} \right) \\
& = & \left( {1 + \frac{1}{2}} \right) - \left( {\frac{1}{n+1} + \frac{1}{{n + 2}}} \right) \\
\end{array}\)
\(\displaystyle \L\lim _{n \to \infty } S_n = \frac{3}{2}\)
S_4 & = & \sum\limits_{k = 1}^4 {\left( {\frac{1}{k} - \frac{1}{{k + 2}}} \right)} \\
& = & \left( {1 - \frac{1}{3}} \right) + \left( {\frac{1}{2} - \frac{1}{4}} \right) + \left( {\frac{1}{3} - \frac{1}{5}} \right) + \left( {\frac{1}{4} - \frac{1}{6}} \right) \\
& = & \left( {1 + \frac{1}{2}} \right) - \left( {\frac{1}{5} + \frac{1}{6}} \right) \\
\end{array}\)
\(\displaystyle \L\begin{array}{rcl}
S_n & = & \sum\limits_{k = 1}^n {\left( {\frac{1}{k} - \frac{1}{{k + 2}}} \right)} \\
& = & \left( {1 - \frac{1}{3}} \right) + \left( {\frac{1}{2} - \frac{1}{4}} \right) + \left( {\frac{1}{3} - \frac{1}{5}} \right) + \cdots + \left( {\frac{1}{n} - \frac{1}{{n + 2}}} \right) \\
& = & \left( {1 + \frac{1}{2}} \right) - \left( {\frac{1}{n+1} + \frac{1}{{n + 2}}} \right) \\
\end{array}\)
\(\displaystyle \L\lim _{n \to \infty } S_n = \frac{3}{2}\)