I'm trying to find the Maclaurin Power Series for $$f(x)=\frac{3x-8}{3x^2+5x-2}$$ but each degree of differentiation gets more complex with no discernible pattern. Any help is appreciated, thanks.
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3Do you know what partial fractions are? Try to break the expression up into simpler ones that you can deal with individually. – Tim May 03 '13 at 21:34
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1@Tim Oh my goodness thank you. I have almost completely forgotten about that. – David May 03 '13 at 21:37
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Partial fractions are good for a lot more than integration. – André Nicolas May 03 '13 at 21:38
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$$\begin{align*} f(x)&=\frac{3x-8}{3x^2+5x-2}=\frac{3x-8}{(x+2)(3x-1)}=\{\text{partial fraction decomposition}\}=\\ &=\frac{2}{2+x}-\frac{3}{3x-1}=\frac{1}{1-\left(-\frac{x}{2}\right)}+\frac{3}{1-3x}=\{\text{geometric series for }|x|<\frac{1}{3}\}=\\ &=\sum_{n=0}^\infty\left(-\frac{x}{2}\right)^n+3\sum_{n=0}^\infty(3x)^n=\sum_{n=0}^\infty\left(\left(-\frac{x}{2}\right)^n+3(3x)^n\right)=\\ &=\sum_{n=0}^\infty\left(\frac{1}{(-2)^n}+3^{n+1}\right)x^n \end{align*}$$
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Thank you for posting that I made an error on the roots and your answer helped me catch it! – David May 03 '13 at 22:35