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

(2.53)

After taking reciprocals of the left‐ and right‐hand sides, Eq. (2.53)gives rise to

(2.54)

or, after recalling Eq. (2.43),

(2.55)

Therefore, the harmonic mean of two numbers never exceeds the geometric mean (again encompassing only positive values), and coincides therewith again when x 1 = x 2.

When n = 2, another mean can be defined as

(2.56)

– known as logarithmic mean; Eq. (2.56)is often rephrased to

(2.57)

after taking advantage of Eq. (2.26). This logarithmic mean lies below the arithmetic and the geometric means, i.e.

(2.58)

where Eq. (2.55)was meanwhile taken advantage of. To prove so, it is convenient to insert Eqs. (2.42), (2.43), (2.45), and (2.57)pertaining to x 1and x 2so as to get

(2.59)

from Eq. (2.58), where factoring of x 1≥ x 2 > 0 (as per working hypothesis) in all sides gives rise to

(2.60)

after dropping x 1 > 0 from all sides, and further multiplying and dividing the last side by x 2/ x 1, Eq. (2.60)turns to

(2.61)

– where z denotes an auxiliary variable satisfying

(2.62)

A graphical account of Eq. (2.60)is provided in Fig. 2.3. Inspection of the curves therein not only unfolds a clear and systematic positioning of the various means relative to each other – in general agreement with (so far, postulated) Eq. (2.58)– but also indicates a collective convergence to x 1as x 2approaches it (as expected).

Figure 2.3 Variation of arithmetic mean ( arm ), logarithmic mean ( lom ), geometric mean ( gem ), and harmonic mean ( ham ) of positive x 1and x 2< x 1, normalized by x 1, as a function of their ratio, x 2/ x 1.

To provide a quantitative argument in support of the graphical trends above, one may expand the middle left‐ and middle right‐hand sides of Eq. (2.61)via Taylor’s series (see discussion later), around z = 1, according to

(2.63)

where 1/2 in the left‐hand side was meanwhile splitted as 1 − 1/2; note that such series in z are convergent because 0 < z < 1, as per Eq. (2.62). Straightforward simplification of Eq. (2.63)unfolds

(2.64)

where z − 1 may be further dropped off both numerator and denominator of the middle side to give

(2.65)

after simplifying notation in Eq. (2.65)to

(2.66)

with the aid of

(2.67)

one may add ζ /2 to all sides to get

(2.68)

– where all terms in the denominator of the middle side are positive, whereas all terms (besides 1) in the right‐hand side are negative. Long (polynomial) division of 1 by 1 + ζ /2 + ζ 2/3 + ζ 3/4 + ⋯ (according to an algorithm to be presented below) allows further transformation of Eq. (2.68)to

(2.69)

where condensation of terms alike in the middle side unfolds

(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).

If u 1, u 2, …, u i, … constitute a given (infinitely long) sequence of numbers, one often needs to calculate the sum of the first n terms thereof – or n th partial sum, S n, defined as