Geometric series of entries of matrix with spectral radius smaller

Question

I have a matrix $A$ such that the spectral radius is $< 1$.

It is well known that $I+A+A^2$... converges. Does it then follow that the geometric series of each entry also converges?

The matrix is non-negatively valued.

Answer 1

No. For example, $A=\pmatrix{0&t\\ 0&0}$ is nilpotent but $1+t+t^2+\cdots$ diverges when $t\ge1$.