Elliptic functions proof -- convergence series on lattice

[binbagsss]

Homework Statement

Hi

I am looking at the proof attached for the theorem attached that:

If $s \in R$, then $\sum'_{w\in\Omega} |w|^-s$ converges iff $s > 2$
where $\Omega \in C$ is a lattice with basis ${w_1,w_2}$.

For any integer $r \geq 0$ :

$\Omega_r := {mw_1+nw_2|m,n \in Z, max {|m|,|n|}=r}$
$\Pi_r := {mw_1+nw_2|m,n \in Z, max {|m|,|n|}=r}$

so that $\Omega = {0} \Cup \Omega_1 \Cup \Omega_2 \Cup...$

Each $\Omega_r$ has cardinality $8r$

QUESTIONS
- To prove via the comparison test, we only need to bound from above by a series that converges, so why have we bound from above and below - this is my main question really, why have we bound from above and below
- Does this proove via both the convergence test and the Weierstass-M test? Since each term in the sequence $|w|^{-s}$ is bound above by a real constant.
- The definition of the W-M test is $u_n$ a seqence of functions, if for each a $n \in N$ there exists $M_n \in R$ satisfying $|u_n(z)|\leq M_n$ got all $z \in E$ where $u_n : E \to C$ and $\sum M_n$ converges. Here the '$u_n$' are $|w|$ are already taken the absolute value, does this change anything here or the W-M test or does it still apply in the same way?

Many thanks in advance.

Homework Equations

as above

The Attempt at a Solution

as above

 

[binbagsss]

bump.many thanks.