1) Given the power series \( \sum_{n=1}^{\infty} \frac{2^{n}}{n}(4 x-8)^{n} \), then: (a) the center of the power series is (b) the radius of convergence of the power series is (c) the interval of convergence of the power series is

Al calcular el \( \lim _{x \mapsto \frac{\Pi}{4}} \frac{\cos x-\sin x}{\cos ^{2} x} \) se obtiene: \( \begin{array}{l}\text { a. } \sqrt{2} \\ \text { b. } 1 \\ \text { c. } 2 / \sqrt{2} \\ \text { d. } \sqrt{2} / 2\end{array} \).

Al calcular el \( \lim _{x \mapsto-\infty} \frac{2 x}{\sqrt{x^{2}+1}} \) se obtiene: \( \begin{array}{l}\text { a. }-2 \\ \text { b. } 1 \\ \text { c. } 2 \\ \text { d. }-1\end{array} \)

\( \lim _ { x \rightarrow \infty } \frac { e ^ { x } \cdot \operatorname { ch } ( 2 x ) } { \operatorname { sh } ( 3 x ) } \)

The cost to produce \( x \) items is modeled by \( C(x)=0.003 x^{2}+3.2 x+60 \) dollars and the revenue from selling \( x \) items is given by \( R(x)=6 x \) dollars. a. Find the marginal profit when 359 items are sold using the formula \( P(x+1)-P(x) \). (Round your answer to three decimal places.) b. Use the derivative to estimate the marginal profit when 359 items are sold. (Round your answer to three decimal places.)

Use polynomial long division to evaluate the integral of \( \frac{x^4 + 6x^3 + 4x^2 + 8}{x^2 + 2} \) for integration.