i.e.,limn→∞vextendsinglevextendsinglevextendsinglevextendsinglean+1anvextendsinglevextendsinglevextendsinglevextendsingle=limn→∞vextendsinglevextendsinglevextendsinglevextendsinglean+1anvextendsinglevextendsinglevextendsinglevextendsingle=limn→∞vextendsinglevextendsinglevextendsinglevextendsingle(−1)n+1(n+ 1)!xn+1(−1)nn!xnvextendsinglevextendsinglevextendsinglevextendsingle=limn→∞(n+ 1)|x|.(5)Forx= 0, this limit is 0 but for anyxnegationslash= 0, the limit is∞. As such, we conclude from the Ratio Testthat the Euler series converges only forx= 0. At allxnegationslash= 0, theseries is divergent.Some remarks on divergent seriesAs discussed briefly in the lecture, series such as the Euler series, which are divergent for allxnegationslash= 0 mayappear to be “totally useless.” That is not the case, however. In fact, divergent series are frequentlyencountered in applied mathematics and theoretical/mathematical physics. In these situations, theyrepresent mathematical functions and can actually be used to reconstruct approximations to thesefunctions. For example, standard “Rayleigh-Schr¨odinger perturbation theory” employed in quantummechanics, e.g., change in energy levels of a hydrogen atom in an electric field (Stark effect) or amagnetic field (Zeeman effect), often yield divergent series.In textbooks, perturbation series wereoften presented only to a couple of terms, with the argument that they were being used to obtainrough estimates of the changes in energies. In some cases, e.g., quantum field theory (at least back inthe 1980’s), perturbation methods represented the only way to understand these theories.