Mathematics for Enzyme Reaction Kinetics and Reactor Performance. F. Xavier Malcata

Eq. (2.84).

Figure 2.4 Variation of value of n‐term arithmetic series, S_n, normalized by first term, u₀, as a function of n – for selected values of increment, k, normalized also by u₀.

      2.1.2 Geometric Series

      A geometric series is said to exist when each term is obtained from the previous one via multiplication by a constant parameter, k, viz.

(2.86)

– and is said to possess ratio k and first term u₀; after factoring u₀ out, Eq. (2.86) becomes

(2.87)

An alternative form of calculating ensues after first rewriting Eq. (2.87) as

(2.88)

      where the first term of the summation was made apparent; multiplication of both sides by k unfolds

      (2.89)

      where straightforward algebraic rearrangement leads to

(2.90)

Ordered subtraction of Eq. (2.90) from Eq. (2.88) generates

      (2.91)

where may be factored out in the left‐hand side and u₀ in its right counterpart to obtain

(2.92)

upon canceling out symmetrical terms in the right‐hand side, and dividing both sides by 1 − k afterward, Eq. (2.92) becomes

(2.93)

Note that k = 1 turns nil both numerator and denominator of Eq. (2.93), so it is a (common) root thereof; Ruffini’s rule (see below) then permits reformulation of Eq. (2.93) to

      (2.94)

      that mimics Eq. (2.87), as expected. Revisiting Eq. (2.72) with division of both sides by u₀, one may insert Eq. (2.93) to get

(2.95)

Eq. (2.95) is illustrated in Fig. 2.5. Upon inspection of this plot, one anticipates a horizontal asymptote for k = 0.5 (besides k = 0); in general, one indeed finds that

      (2.96)

      at the expense of Eqs. (2.72), (2.73), and (2.95), which is equivalent to

(2.97)

– and reduces to merely

(2.98)

      should k lie between −1 and 1, as it would give rise to a convergent series (i.e. k^{n + 1} → 0 when n → ∞, in this case). All remaining values, i.e. k ≤ −1 or k ≥ 1, produce divergent series – as apparent in Fig. 2.5; when k is nil, Eq. (2.93) turns to just

      (2.99)

which corresponds to the trivial case of only the first term of said series being significant – and fully consistent with Eqs. (2.97) and (2.98).

Figure 2.5 Variation of value of n‐term geometric series, S_n, normalized by first term, u₀, as a function of n – for selected values of ratio, k.

      An alternative proof of Eq. (2.93) comes from finite induction; one should first confirm that it applies to n = 0, i.e.

      (2.100)