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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 3

$\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \frac{4}{5!} + \ldots = \ ?$

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

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

$\frac{1}{(x-1)(x-2)} + \frac{1}{(x-2)(x-3)} + \frac{1}{(x-3)(x-4)} = \frac{1}{6}$

What is the sum of all real values of $$x$$ that satisfy the above equation?

$\LARGE \sqrt[3]{ \sqrt{32}} \sqrt[4]{ \sqrt[3]{32}} \sqrt[5]{ \sqrt[4]{32}} \cdots \sqrt[10]{ \sqrt[9]{32}} = \ ?$

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