Trying to show that logarithm of this matrix does not convergence let $A \in O(3) - SO(3)$, i.e, $A$ is an orthogonal  $3 \times 3$ matrix with real entries that has a determinant $-1$. I'm trying to show that $logA$ defined as:
\begin{equation*}
logA = (A-I) - \frac{(A - I)^2}{2} + \frac{(A - I)^3}{3} - \frac{(A - I)^4}{4} + ...
\end{equation*}
Does not convergence.
My thinking: We know that $A$ has to be close to $I$, the identity matrix, for $logA$ to convergence. Since $I \in SO(3)$, i.e, it is an orthogonal matrix with determinant $1$, and $A \in O(3) - SO(3)$, it can not be contained in a small neighborhood of $I$ thus cannot converge.
However, I couldn't show this in any rigorous way. Any help would be appreciated.
Thanks.

Let's break this down.  If a matrix $A$ has determinant $d$, then $\log A$ will have determinant $\log d$.   Can we get a sum with determinant $\log d$ from a series of real matrices when $d=-1$?

Yup, it's bad.
