求和

未分类

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$$\begin{aligned} & \left( {1\over{1\times 2007}} + {1\over{2\times 2006}} + \dots + {1\over{2007\times 1}} \right) - {2007\over 2008} \times \left( {1\over{1\times 2006}} + {1\over{2\times 2005}} + \dots + {1\over{2006\times 1}} \right)\\\\\\\\ =& \begin{aligned}{1\over 2008}\times &\left( {2008\over{1\times 2007}} + {2008\over{2\times 2006}} + \dots + {2008\over{2007\times 1}} \right) \\ & - {2007\over 2008} \times \left( {1\over{1\times 2006}} + {1\over{2\times 2005}} + \dots + {1\over{2006\times 1}} \right)\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left[\left( {2008\over{1\times 2007}} + {2008\over{2\times 2006}} + \dots + {2008\over{2007\times 1}} \right) - \\{2007} \times \left( {1\over{1\times 2006}} + {1\over{2\times 2005}} + \dots + {1\over{2006\times 1}} \right)\right]\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left[\left( {2008\over{1\times 2007}} + {2008\over{2\times 2006}} + \dots + {2008\over{2006\times 2}} + {2008\over{2007\times 1}} \right)\\ - \left( {2007\over{1\times 2006}}- + {2007\over{2\times 2005}} + \dots + {2007\over{2006\times 1}} \right)\right]\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left[\left( {2008\over{1\times 2007}} - {2007\over{1\times 2006}}\right) + \left({2008\over{2\times 2006}} - {2007\over{2\times 2005}}\right) + \\\left({2008\over{3\times 2005}} - {2007\over{3\times 2004}}\right) \dots + \left({2008\over{2006\times 2}} - {2007\over{2006\times 1}}\right) + {2008\over{2007\times 1}}\right]\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left({ {2008 \times 2006 - 2007 \times 2007}\over{1\times 2007\times 2006}} + { {2008 \times 2005 - 2007 \times 2006}\over{2\times 2006\times 2005}} + \\{ {2008 \times 2004 - 2007 \times 2005}\over{3\times 2005\times 2004}} \dots + {{2008 \times 1 - 2007 \times 2}\over{2006\times 2\times 1}} + {2008\over{2007\times 1}}\right)\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left[{ {(2007+1)\times(2007-1) - 2007 \times 2007}\over{1\times 2007\times 2006}} + \\{ {(2007+1) \times (2006-1) - 2007 \times 2006}\over{2\times 2006\times 2005}} + \\{ {(2007+1) \times (2005-1) - 2007 \times 2005}\over{3\times 2005\times 2004}} + \\\dots + {{(2007+1) \times (2-1) - 2007 \times 2}\over{2006\times 2\times 1}} + {2008\over{2007\times 1}}\right]\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left({ {-1}\over{1\times 2007\times 2006}} + { {-2}\over{2\times 2006\times 2005}} + \\{ {-3}\over{3\times 2005\times 2004}} + \dots + {{-2006}\over{2006\times 2\times 1}} + {2008\over{2007\times 1}}\right)\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left({ {-1}\over{2007\times 2006}} + { {-1}\over{2006\times 2005}} + \\{ {-1}\over{2005\times 2004}} + \dots + {{-1}\over{2\times 1}} + {2008\over{2007\times 1}}\right)\end{aligned}\\\\\\\\ =& \begin{aligned}{1\over 2008}\times \left[\left({1 \over 2007} - {1 \over 2006}\right) + \left({1 \over 2006} - {1 \over 2005}\right) + \\ \left({1 \over 2005} - {1 \over 2004}\right) + \dots + \left({1\over 2} - {1\over 1}\right) + {2008\over{2007\times 1}}\right]\end{aligned}\\\\\\\\ =& {1\over 2008}\times \left[{1 \over 2007} - {1 \over 2006} + {1 \over 2006} - {1 \over 2005} + \\ {1 \over 2005} - {1 \over 2004} + \dots + {1\over 2} - {1\over 1} + {2008\over{2007\times 1}}\right]\\\\\\\\ =& {1\over 2008}\times \left({1 \over 2007} - {1\over 1} + {2008\over{2007\times 1}}\right)\\\\ \end{aligned}$$

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