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

(2.72)

and also known as series; if the partial sums S 1, S 2, …, 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.

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 u 0. Equation (2.76)may obviously be reformulated to

(2.77)

using the last term,

(2.78)

instead of the first one, u 0, 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 u 0, 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 u 0and 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 nwith n , in agreement with Eq. (2.84).

Figure 2.4 Variation of value of n ‐term arithmetic series, S n, normalized by first term, u 0, as a function of n – for selected values of increment, k , normalized also by u 0.

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 u 0; after factoring u 0out, 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)