  We prove that the linear delta expansion for energy eigenvalues of the
quantum mechanical anharmonic oscillator converges to the exact answer if the
order dependent trial frequency $\Omega$ is chosen to scale with the order as
$\Omega=CN^\gamma$; $1/3<\gamma<1/2$, $C>0$ as $N\rightarrow\infty$. It
converges also for $\gamma=1/3$, if $C\geq\alpha_c g^{1/3}$, $\alpha_c\simeq
0.570875$, where $g$ is the coupling constant in front of the operator $q^4/4$.
The extreme case with $\gamma=1/3$, $C=\alpha_cg^{1/3}$ corresponds to the
choice discussed earlier by Seznec and Zinn-Justin and, more recently, by
Duncan and Jones.
