Is it correct to use a comma in this equation?

Question

Consider:

Then, the sequence $k_i$ given by (3.38) is increasing and converges to 
\begin{equation}\label{eq:weired_equation}
k^\xi = k_1\times k_2 \times c \times d,
\end{equation}
where $c$ and $d$ are defined in the Theorem 2. 

I have the questions:

1. What are the grammatical errors in the above piece?
2. Should a comma be used after the equation \eqref{eq:weired_eqution}.

Answer 1

There should be no difference in punctuation whether you write the equation on display or inside the text.

So since there would be a comma if written inside the text, the comma is correct.

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Considering the 1st question, see my comment. In the end, the piece can look like

Then the sequence $k_i$ given by (3.38) is increasing and converges to 
\begin{equation}\label{eq:weired_equation}
k^\xi = k_1\times k_2 \times c \times d,
\end{equation}
where $c$ and $d$ are defined in Theorem 2. 

In the label, there's a typo and it should be weird_equation, but I did not correct this to avoid you problems with cross-referencing.