A new series for the phase shift has been derived for the Schrödinger, Klein-Gordon, and Dirac equations. This series converges faster than the Born series for the tangent of the phase shift. This is so because the sum of the first $n$ terms in the new series includes exactly all the terms up to the $2(2^n-1)\mathrm{th}$ order in the Born series. Under the condition which is tantamount to that the phase shift cannot be larger than 63°, the series converges absolutely. At high energies the series can be analytically continued with respect to the strength of the potential beyond such a limit. It is shown that the high-energy limit of the phase shift is given by its first Born approximation and that the difference between even and noneven potentials is reflected in the respective phase shifts to all orders.

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