is made available in Fig. 2.6 – after recalling Eq. (2.72) and dividing both sides by u0, i.e.
n, k1/u0 and k2 were consequently utilized as independent parameters. An increasing n systematically produces a higher‐value series – and a similar effect is triggered by a larger k1 or a larger k2; there is a horizontal asymptote when k2 = 0.5 (besides the trivial case of k2 = 0), in much the same way horizontal asymptotes arose in Fig. 2.5.
Figure 2.6 Variation of value of n‐term arithmetic–geometric series, Sn, normalized by first term, u0, as a function of n – for selected values of increment, k1, normalized also by u0, and ratio, k2, for (a) k1/u0 = 0, (b) k1/u0 = 0.5, (c) k1/u0 = 1, and (d) k1/u0 = 2.
In general, one realizes that
(2.118)
with the aid of Eqs. (2.72), (2.73), and (2.117), or else
after direct application of the theorems on limits. If ∣k2∣ < 1, then Eq. (2.119) degenerates to
(2.120)
– where the first term entails an unknown quantity, while
(2.121)
one gets an unknown quantity of the type ∞/∞ – so one may apply l’Hôpital’s rule to get
Equation (2.122) degenerates to
since ∣k2 ∣ < 1; insertion of Eq. (2.123) allows final transformation of Eq. (2.119) to
which describes the vertical intercept of the horizontal asymptotes in Fig. 2.6 for curves with
The trivial case, associated with k1 = 0, transforms indeed Eq. (2.116) to
after having u0 factored out – and serves as descriptor of the curves plotted in Fig. 2.6 a; Eq. (2.125) coincides with Eq. (2.93), corresponding to a plain geometric series, as expected from
(2.126)
that in turn stems from Eqs. (2.87) and (2.110) – thus fully justifying coincidence between Figs. 2.5 and 2.6a. Conversely, k2 = 0 converts Eq. (2.116) to
(2.127)
which serves as descriptor of the bottom curves in Fig. 2.6, labeled as k2 = 0. When k2 → 1, Eq. (2.116) becomes
if the theorems on limits are blindly applied. To circumvent the unknown quantities of the 0/0 type, one may independently differentiate, with regard to k2, numerator and denominator of either term in the right‐hand side of Eq.
