  A classical observation of Riesz says that truncations of a general
$\sum_{n=0}^\infty a_n z^n$ in the Hardy space $H^1$ do not converge in $H^1$.
A substitute positive result is proved: these partial sums always converge in
the Bergman norm $A^1$. The result is extended to complete Reinhardt domains in
$\C^n$. A new proof of the failure of $H^1$ convergence is also given.
