= Zero-dimensional quartic perturbation series
{title2=$I(\lambda)/I(0)=1-\lambda/(8m^4)+35\lambda^2/(384m^8)+\cdots$}
For $m^2>0$, the <zero-dimensional scalar field theory> with <Euclidean action> $m^2x^2/2+\lambda x^4/4!$ has a <Gaussian integral> normalization $I(0)=\sqrt{2\pi/m^2}$. Its <Feynman rules> assign $1/m^2$ to each <propagator>, $-\lambda$ to each quartic vertex, and the reciprocal <Feynman-diagram symmetry factor> to each <vacuum diagram>. The integral itself includes disconnected <vacuum diagrams>; its logarithm selects connected ones. At second order the three possible numbers of edges joining the two vertices are zero, two and four, with <Feynman-diagram symmetry factors> $128,16,48$ respectively. Their weights add to $35\lambda^2/(384m^8)$.
The <Wick contractions> give the general coefficient $I(0)(-1)^v(4v-1)!!/[v!(4!)^v m^{4v}]$. Successive absolute coefficients have ratio $(4v+3)(4v+1)/[24(v+1)m^4]$, which tends to infinity. Thus this is an <asymptotic expansion> as $\lambda\downarrow0$, rather than a convergent <power series>. For every fixed truncation the <Taylor theorem> remainder estimate for $e^{-u}$, $u\geq0$, can be integrated against the <Gaussian distribution> to justify the corresponding remainder bound.
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