divergence test for series (also called the " \( n^{\text {th }} \) term test for divergence of a series" \[ \lim _{n \rightarrow \infty} a_{n} \neq 0 \Rightarrow \sum_{n=1}^{\infty} a_{n} \text { diverges } \] verge or it might diverge. sider the series \( \sum_{n=1}^{\infty} \frac{5 n^{4}}{2 n^{2}+6} \). mighat this test tells us nothing about \( \sum_{n=1}^{\infty} a_{n} \) if \( \lim _{n \rightarrow \infty} a_{n}=0 \); in that situation th mongence test tells us that this series: converges diverges

18. La integral de \( \int \frac{d x}{x+5}=e s: \) \( \begin{array}{ll}\text { a) } \frac{\ln |x+5|}{-5}+c & \text { b) } \frac{\ln 3}{-x}+c \\ \text { c) } \ln |x+5|+c & \text { d) } \frac{x \ln e^{-2}}{-2}+c \\ \text { A) } & \\ \text { B) } \\ \text { C) } \\ \text { D) }\end{array} \)

Let \( R \) be the region enclosed by the graphs of \( f(x)=4 \cos (x)-1 \) and \( g(x)=4 \sin (x)-2 \). Find the length of the boundary of region \( R \). Use a calculator and round your answer to 3 decimal places.

Dada la ecuación diferencial \( y' = -4y + 12 \), aplica el método de Euler desde \( x=0 \) hasta \( x=0.5 \) con un paso de \( h=0.1 \).

1) \( \int x \sqrt[3]{x^{4}+2} d x \)

17. La fintegralde \( \int 3 e^{3 x} d x=e s: \) \( \begin{array}{ll}\text { a) } \frac{e^{3 x}}{-2}+c & \text { b) } \frac{3}{-x}+c \\ \text { c) } e^{3 x}+c & \text { d) } \frac{x e^{-2}}{-2}+c\end{array} \)