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

(2.70)

after having dropped unity from all sides, and then taken their negatives, Eq. (2.70) becomes

(2.71)

– which is a universal condition, since 1/12 < 1/8, 1/24 < 1/16, and so on in terms of pairwise comparison. Similar trends for the relative magnitude of the coefficients of similar powers would be found if the series were truncated after higher order terms – so one concludes on the general validity of Eq. (2.58), based on Eq. (2.71) complemented by Eq. (2.55).

2.1 Series

      If u₁, u₂, …, u_i, … constitute a given (infinitely long) sequence of numbers, one often needs to calculate the sum of the first n terms thereof – or nth partial sum, S_n, defined as

(2.72)

      and also known as series; if the partial sums S₁, S₂, …, S_i, … converge to a finite limit, say, S, according to

(2.73)

      then S can be viewed as the infinite series

      (2.74)

      – while the said series is termed convergent. Should the sequence of partial sums tend to infinite, or oscillate either finitely or infinitely, then the series would be termed divergent.

      Despite the great many series that may be devised, two of them possess major practical importance – arithmetic and geometric progressions, as well as their hybrid (i.e. arithmetic–geometric progressions); hence, all three types will be treated below in detail.

      2.1.1 Arithmetic Series

      Consider a series with n terms, where each term, u_i, equals the previous one, u_i–1, added to a constant value, k – according to

(2.75)

      or, equivalently,

(2.76)

is termed a (finite) arithmetic progression, or arithmetic series, of increment k and first term u₀. Equation (2.76) may obviously be reformulated to

(2.77)

      using the last term,

(2.78)

instead of the first one, u₀, as reference; upon ordered addition of Eqs. (2.75) and (2.77), one obtains

(2.79)

– where cancelation of symmetrical terms reduces Eq. (2.79) to

(2.80)

Upon factoring n + 1 out in the right‐hand side, followed by division of both sides by 2, Eq. (2.80) becomes

      (2.81)

– i.e. it looks as n + 1 times the arithmetic average of the first and last terms of the series; insertion of Eq. (2.78) permits transformation to

      (2.82)

      that breaks down to

(2.83)

      – valid irrespective of the actual values of u₀, k or n.

Note that an arithmetic series is never convergent in the sense put forward by Eqs. (2.72) and (2.73), because the magnitude of each individual term keeps increasing without bound as n → ∞; this becomes apparent after dividing both sides of Eq. (2.83) by u₀ and retrieving Eq. (2.72), i.e.

(2.84)

which translates to Fig. 2.4. In the particular case of k = 0, Eq. (2.83) simplifies to

      (2.85)

consistent with the definition of multiplication – which describes the lowest curve in Fig. 2.4, essentially materialized by a straight line with unit slope for relatively large n; as expected, a large k eventually produces a quadratic growth of S_n