Prediction of unknown terms of a sequence and its application to some physical problems

Nonlinear sequence transforms are known to predict the limits of a sequence, convergent or divergent. The power of these nonlinear transforms, especially the Levin-like transforms in predicting the subsequent terms of a sequence or series is investigated. If a particular sequence transform can predict one extra term of a series or sequence, then by including that term one can build a higher order transform and predict the subsequent term. Repeating the procedure one can, in principle, predict all the terms of the sequence and, in this sense, a complete knowledge of the sequence can be obtained. An analysis of the Levin-like transforms reveals that this is possible for a large number of test sequences, even though these transforms do not sum them exactly. We then apply this method to the divergent power series involved in a number of diverse physical problems, namely the ground state energy of the hydrogen atom in a magnetic field (the Zeeman problem), the ground state energy of the quartic anharmonic oscillator, the expansion of thermodynamic properties of a solid in terms of the moments of the frequency spectrum, the Ising model and the excluded volume effect in polymers. We also apply the method to problems in quantum electrodynamics (QED), namely to the perturbation series for the anomalous magnetic moment of the electron and the muon and to the ratios R (e⁺e⁻) = σ_tot (e⁺e⁻ → hadrons) /σ (e⁺e⁻ → μ⁺μ⁻) and which have played instrumental roles in the development of quantum chromodynamics (QCD). In all these cases it is observed that the Levin-like transforms can be used to predict the subsequent terms of the sequences with high accuracy and these predicted terms can be included to calculate the physical properties with a greater accuracy by using higher order transforms. Thus, such predictions are expected to be very helpful in situations where the evaluation of the higher order terms of a sequence or series becomes extremely difficult as is the case in many physical situations.