En los ejercicios 1 a 8 encuentre la serie de Maclaurin (es decir, serie de Taylor alrededor de \( c=0 \) ) y su intervalo de convergencia 1. \( f(x)=\cos x \) 2. \( f(x)=\operatorname{sen} x \) 3. \( f(x)=e^{2 x} \) 4. \( f(x)=\cos 2 x \) 5. \( f(x)=\ln (1+x) \) 6. \( f(x)=e^{-x} \) 7. \( f(x)=1 /(1+x)^{2} \) 8. \( f(x)=1 /(1-x) \)
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Step 1: **Find the Maclaurin series for \( f(x) = \cos x \)** - The Maclaurin series for \( \cos x \) is given by: \[ \cos x = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n}}{(2n)!} \] - Interval of convergence: \( (-\infty, \infty) \) Show more…
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