<s>[INST] How do I compute the taylor series for $(1+x+x^2)^\frac{1}{x}$ I tried rewriting $(1+x+x^2)^\frac{1}{x}$ as $e^{\frac{1}{x}\ln(1+x+x^2)}$ and then computing the taylor series of $\frac{1}{x}$ and $\ln(1+x+x^2)$ but I'm still not getting the correct answer..
 [/INST] Note that $$f(x) = (1 + x + x^{2})^{1/x} = \exp\left(\frac{\log(1 + x + x^{2})}{x}\right) = \exp\left(\frac{\log(1 - x^{3}) - \log(1 - x)}{x}\right)$$ so that we can get the series $$\frac{\log(1 - x^{3}) - \log(1 - x)}{x} = \sum_{n = 1}^{\infty}\frac{x^{n - 1} - x^{3n - 1}}{n} = 1 + \frac{x}{2} - \frac{2x^{2}}{3} + \cdots$$ and we can exponentiate this to get $$(1 + x + x^{2})^{1/x} = e\cdot\exp(x/2 - 2x^{2}/3 + \cdots) = e\left(1 + \frac{x}{2} - \frac{13}{24}x^{2} + \cdots\right)$$ Note that the calculation of coefficients in general is complicated with no specific formula.
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