Convergence of the Jacobi iteration method I think I am not quite understanding the Jacobi Method or some related concept for indirectly solving linear systems of equations of the form $Ax=b$. We need the norm
$||I-Q^{-1}A||_\infty < 1$ and the Jacobi Method states to let $Q$ be the matrix with the same diagonal entries as $A$. Then we have:
\begin{equation}                                                                                                                                                            
||I-Q^{-1}A||_\infty = \max_{1\leq i\leq n}\sum\limits_{j=1,j\neq i}^n \left|\frac{a_{ij}}{a_{ii}}\right|                                                                   
\end{equation}
Let $A$ be a diagonally dominant matrix. We see that $Q^{-1}A$ has $1$'s on the diagonal. the matrix $I-Q^{-1}A$ has $0$s on the diagonal and the sum of each row is less
than 1. My book then concludes that $||I-Q^{-1}A||_\infty < 1$. It is not obvious to me from the above equations that this is true. I think I constructed a
counter-example:
\begin{align}                                                                                                                                                               
A=&\begin{pmatrix}\frac{1}{2}&\frac{1}{4}&\frac{1}{8}\\                                                                                                                     
                 0&\frac{2}{3}&\frac{1}{2}\\                                                                                                                                
                 3&\frac{1}{2}&4\end{pmatrix}\\                                                                                                                             
Q^{-1}A =& \begin{pmatrix}1&\frac{1}{2}&\frac{1}{4}\\                                                                                                                       
                          0&1&\frac{3}{4}\\                                                                                                                                 
                          \frac{3}{4}&\frac{1}{8}&1\end{pmatrix}\\                                                                                                          
I-Q^{-1}A =& \begin{pmatrix}0&\frac{-1}{2}&\frac{-1}{4}\\                                                                                                                     
                          0&0&\frac{-3}{4}\\                                                                                                                                 
                          \frac{-3}{4}&\frac{-1}{8}&0\end{pmatrix}                                                                                                            
\end{align}
$A$ is diagonally dominant but $||I-Q^{-1}A||_\infty$ which is the sum of the absolute value of the maximum entry in each column $=\frac{3}{4}+\frac{1}{2}+\frac{3}{4}>1$
Is this correct? If not, what did I do wrong?

Your understanding of the infinity norm for matrices is wrong:
$$ \Vert I - Q^{-1}A \Vert_\infty = \max \bigg \{0+\bigg|-\frac 12\bigg| + \bigg|-\frac14\bigg|, 0 +0+\bigg|-\frac34\bigg|, \bigg|-\frac34\bigg| + \bigg|-\frac18\bigg| + 0 \bigg\} = \frac 7 8 < 1$$
