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Proof of the convergence of the linear delta expansion: Zero dimensions.

The convergence of the linear δ expansion is studied in the context of the integral I: = ∫ − ∞ ∞e −gx4 dx, which corresponds to massless φ 4 theory in 0 dimensions. The method consists of rewriting the exponent as −δgx 4 −λ(1−δ)x 2 and expanding in powers of δ. The arbitrary parameter λ is fixed by the principle of minimal sensitivity, ∂I K (λ)/∂λ = 0, where I K is the expansion truncated at order K with δ set equal to 1. This has a solution λ 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 I K (λ K ) is convergent, the error decreasing exponentially with K, even though for fixed λ the series expansion is a divergent alternating series