$(a)$ Test for the convergence $or$ divergence $of$ the following series:

$(i){\displaystyle \underset{n=1}{\overset{}{}}}(1+\frac{3}{n}{)}^{n}by$ first deriving the $\underset{n+}{lim}(1+\frac{3}{n}{)}^n.$

$(ii){\displaystyle \underset{n=1}{\overset{}{}}}\frac{n^2}{(2n-1)!}.$

$3(a)$ Consider the following integral:

$I={}_{}\frac{(2x-y{)}^2}{2x+y}dxdy$

where $is$ the region bounded $byx=0,y=0,y=1-2x$ and $y=2-2x.$

$(i)$ Use the transformation

$x=\frac{1}{4}(u-v),$ and $,y=\frac{1}{2}(u+v)$

$to$ sketch the area $of$ integration $in$ the $uv-$plane.

$(ii)$ Calculate the Jacobian $of$ transformation $|\frac{\text{d}\text{e}\text{l}(x,y)}{\text{d}\text{e}\text{l}(u,v)}|.$

$(iii)$ Hence, use the transformation $(1)to$ evaluate the integral $1.$

$[4+2+5$ marks$]$

$(b)$ Let $Vbe$ the volume bounded $by$ the sphere $x^2+y^2+z^2=25$ and inside the

cone $z^2=x^2+y^2$ above the plane $z=0.$ Describe the region $of$ integration

representing $Vby$ using the spherical coordinates. Hence, calculate $Vby$

writing $itin$ the form $k(a-\sqrt[\phantom{2}]{2}),$ where $k$ and a are $tobe$ determined.

$[7$ marks$]$

$(c)$ Assume that the vector space $R^3$ has the Euclidean inner product. Apply the

Gram$-$Schmidt algorithm $to$ orthonormalise the set $of$ vectors

vec$(u{)}_1=(-1,1,1{)}^T,$vec$(u{)}_2=(0,1,1{)}^T,$ and $,$vec$(u{)}_3=(1,2,1{)}^T.$