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- \documentclass{article}
- \usepackage{amsmath}
- \usepackage{newpxtext}
- \usepackage{eulervm}
- \pagestyle{empty}
- \begin{document}
- \[
- \everymath={\displaystyle}
- \def\arraystretch{3.0}
- \begin{array}{r@{\ }l@{\ }l}
- \sum_{n = 0}^\infty F_n x^n
- & = & \frac{-x}{x^2 + x - 1} \\
- & = & \frac{-x}{\bigl( x - \frac{1}{\phi} \bigr) \bigl( x + \phi \bigr)} \\
- & = & \frac{-1}{\sqrt{5}} \left( \frac{\frac{1}{\phi}}{x - \frac{1}{\phi}} + \frac{\phi}{x + \phi} \right) \\
- & = & \frac{1}{\sqrt{5}} \left( \frac{1}{1 - \phi x} - \frac{1}{1 + \frac{1}{\phi} x} \right) \\
- & = & \frac{1}{\sqrt{5}} \left[ \sum_{n = 0}^\infty \phi^n x^n - \sum_{n = 0}^\infty \left( -\frac{1}{\phi} \right)^n x^n \right] \\
- & = & \sum_{n = 0}^\infty \frac{\phi^n - \bigl( -\frac{1}{\phi} \bigr)^n}{\sqrt{5}} x^n
- \\[20pt]
- F_n
- & = & \frac{1}{\sqrt{5}} \left[ \left( \frac{1 + \sqrt{5}}{2} \right)^n - \left( \frac{1 - \sqrt{5}}{2} \right)^n \right]
- \end{array}
- \]
- \end{document}
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