$(1)(15pnts)$ Let $F(x,y,z)=(y{e}^x+e^z)i+e^{x}j+x{e}^{z}kbe$ a vector field.

$(a)$ Find curl$(F)$ and div$(F).$

$(b)IsF$ a conservative vector field? $If$ yes, then find potential func$-$

tion.

$(2)(10pnts)$ Find the counterclockwise circulation $(work)ofF(x,y)=$

$(y+e^x{x}^3)i+(2x-e^y)j$ around the boundary $of$ the region $is$ bounded

above $by$ the curve $y=3-x^2$ and below $by$ the curve $y=x^2+1$

$(Hint$: Use Green Theorem$).$

$(3)(50pnts)$ Determine whether $if$ the following series are convergent,

divergent, conditionally convergent $orabsolutely$ convergent.

$(a){\displaystyle \underset{n=1}{\overset{}{}}}(-1{)}^n\frac{n^2+1}{2n^3+n+1}$

$(b){\displaystyle \underset{n=0}{\overset{}{}}}\frac{2^n}{e^{n+1}}$

$(c){\displaystyle \underset{n=1}{\overset{}{}}}\frac{(-1{)}^n}{n+nl{n}^{2}n}$

$(d){\displaystyle \underset{n=1}{\overset{}{}}}(\frac{n}{3n+1}{)}^n$

$(e){\displaystyle \underset{n=1}{\overset{}{}}}(tan(\frac{1}{n})-tan(\frac{1}{n+1}))$

$(4)(15pnts)$ Find the power series representation $of$ the given function

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{x}^{3}arctan(x^2)dx$

$(5)(10)$ Find the interval $of$ convergence and radius $of$ convergence for the

power series ${\displaystyle \underset{n=1}{\overset{}{}}}\frac{n(x+3{)}^n}{5^n}$