Читать книгу Mathematics for Enzyme Reaction Kinetics and Reactor Performance - F. Xavier Malcata - Страница 19

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 with n, in agreement with 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.