Ch. 4 considers the regularisation of some basic divergent series. The first is the
geometric series, which is shown to be conditionally convergent outside the unit circle of absolute
convergence forℜz<1 and divergent elsewhere. Nevertheless, the regularised value is found to
be identical to the limit of 1/(1−z)when the series is convergent. Then the regularisation of the
binomial series is considered, where again, the regularised value is found to equal the limit when
the series is convergent. Next the series denoted by
2F1
(a+1,b+1;a+b+2−x; 1), which is
divergent forℜx>0, is analysed. Here the regularised value is found to be different from the
limit when the series is convergent or for ℜx<0. Finally, the harmonic series is studied, whose
regularised value equals Euler’s constant. Unlike the previous examples, the last example involves
logarithmic regularisation.
