What are the tensor and series questions in this homework?

[Pual Black]

Homework Statement

i have a few homework question and want to be sure if I have solved them right.
Q1) Write $\vec{\triangledown}\cdot\vec{\triangledown}\times\vec{A}$ and $\vec{\triangledown}\times\vec{\triangledown}\phi$ in tensor index notation in $R^3$

Q2) the spherical coordinates
$x=r sin\theta cos\phi$
$y=r sin\theta sin\phi$
$z=r cos\theta$
what is the relataion of $dx, dy, dz$ in terms of $dr , d\theta , d\phi ,$

Q3) Determine whether the following series converges
$\sum \left(\frac{2}{5^{k+1} }+\frac{(2k)!}{3^k}\right)$
this problem has no summation startpoint. I thought such question must have a start point and go to infinity. like k=0 or k=2

The Attempt at a Solution

Q1) $\vec{\triangledown}\cdot\vec{\triangledown}\times\vec{A} = \epsilon_{ijk}\partial_{i}\partial_{j}A_{k}$
$\vec{\triangledown}\times\vec{\triangledown}\phi = \epsilon_{ijk}\partial_{j}\partial_{k}\phi$

Q2) $dx=sin\theta cos\phi dr + r cos\theta cos\phi d\theta - r sin\theta sin\phi dphi$
$dy=sin\theta sin\phi dr + r cos\theta sin\phi d\phi + r cos\theta cos\phi d\phi$
$dz=cos\theta dr - r sin\theta d\theta$

Q3)
$\sum \frac{2}{5^{k+1} }=\frac{2}{5}+\frac{2}{25}+\frac{2}{125}+...$
from geometric series
$\lim_{k \rightarrow \infty}S_n=\frac{a}{1-r}$

$a=\frac{2}{5}$ $r=\frac{1}{5}$

since $\mid r\mid<1$ the series converges
$\lim_{k \rightarrow \infty}S_n=\frac{\frac{2}{5}}{1-\frac{1}{5}}=\frac{1}{2}$

$\sum \frac{(2k)!}{3^k}$

using D'Alembert ratio test
$\rho=\lim_{k \rightarrow \infty} \frac{u_{k+1}}{u_{k}}$

$\rho=\lim_{k \rightarrow \infty} \frac{\frac{[2(k+1)]!}{3^{k+1}}}{\frac{(2k)!}{3^{k}}}$$\rho=\lim_{k \rightarrow \infty} \frac{(2k+2)!3^{k}}{(2k)!3^{k+1}}$$\rho=\lim_{k \rightarrow \infty} \frac{(2k+2)(2k+1)}{3}$

this gives infinity and therefore this series diverges

 

[Orodruin]

Q1) The first expression is correct. The second is missing a basis vector and has a mismatched free index on the right hand side.

Q2) Double check the last term of dy.

Q3) ok

 

[Pual Black]

Q1) The first expression is correct. The second is missing a basis vector and has a mismatched free index on the right hand side.

Q2) Double check the last term of dy.

Q3) ok

Q1) the second expression. I think $\phi$ is a scalar and therefore it should not have an index. Right?

Q2) yes i made a mistak. It should be
$dy=sin\theta sin\phi dr + r cos\theta sin\phi d\theta + r sin\theta cos\phi d\phi$

Q3) so the final answer is
Converge + Diverge = Diverge.
And is it normal that the Summation has no startpoint? I thought there is a trick.