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Partial Fractions

Express rational functions as a sum of fractions with simpler denominators. You can apply this to telescoping series to mass cancel terms in a seemingly complicated sum.

Level 2

         

\[\large \displaystyle \sum_{n = 1}^{\infty} \dfrac {1}{n^2 + 3n + 2} = \ ? \]

\[1 + \frac {1}{\color{blue}2} + \frac {1}{\color{blue}6} + \frac {1}{\color{blue}{12}} + \frac {1}{\color{blue}{20}} + \ldots = \ \color{teal}? \]

Consider the following pattern:

\[\begin{align} \frac{1}{1\times 2} & = \frac{1}{1} - \frac{1}{2} \\ & \\ \frac{1}{2\times 3} & = \frac{1}{2} - \frac{1}{3} \\ & \\ \frac{1}{3\times 4} & = \frac{1}{3} - \frac{1}{4} \\ \vdots & \end{align} \]

Following the pattern above, if \( \displaystyle \frac{1}{11\times 12} = \frac{1}{a} - \frac{1}{b}, \) what are the values of \( a \) and \( b\)?

\[ \large \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\cdots +\frac{1}{99 \times 100} = \ ? \]

\[\large \frac{1}{1\cdot 3} + \frac{1}{3\cdot 5} +\frac{1}{5\cdot 7} + \frac{1}{7 \cdot 9} + \cdots = \ ?\]

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