Proof of the convergence of the linear δ expansion: Zero dimensions

The convergence of the linear δ expansion is studied in the context of the integral I:=${\mathcal{F}}_{\mathrm{-}\mathrm{\infty }}^{\mathrm{\infty }}$${\mathit{e}}^{\mathrm{-}\mathit{g}\mathit{x}4}$dx, which corresponds to massless ${\mathit{cphi}}^4$ theory in 0 dimensions. The method consists of rewriting the exponent as -δ${\mathit{gx}}^4$-λ(1-δ)${\mathit{x}}^2$ and expanding in powers of δ. The arbitrary parameter λ is fixed by the principle of minimal sensitivity, ∂${\mathit{I}}_{\mathit{K}}$(λ)/∂λ=0, where ${\mathit{I}}_{\mathit{K}}$ is the expansion truncated at order K with δ set equal to 1. This has a solution λ${\mathrm{¯}}_{\mathit{K}}$ only for K odd, when it gives very good numerical results. We are able to show analytically, using saddle-point methods, that the sequence of approximants ${\mathit{I}}_{\mathit{K}}$(λ${\mathrm{¯}}_{\mathit{K}}$) is convergent, the error decreasing exponentially with K, even though for fixed λ the series expansion is a divergent alternating series.