156
II - Convergence: Discrete variables
(19.14)
(1 - z)-P = LP(p + 1) ... (p + n - l)z[n J •
On putting -p = s and replacing z by -z, we get
(19.15)
(1 + z)S = L s(s - 1) ... (s - n + l)z[n J •
If s were a positive integer the coefficient of zn would be zero for n ~ s+ 1 since
it would contain s - s = ° as a factor; the preceding formula would reduce
to the algebraic binomial formula recalled in n° 1. The result obtained shows
that, for s a negative integer, it remains valid on condition:
(i) one writes its coefficients in the form above, and not in the form
s!/n!(s-n)!: this is meaningless apart from the case where sEN and Newton,
who very happily did not know this - it would have led him down a blind alley
-, never did so, contenting himself with the form (15) which has a meaning
for any SEC,
(ii) one replaces the finite sum of algebra by an infinite series,
(iii) one does not forget that the formula obtained is valid only for JzJ < 1,
since otherwise the series is divergent as we saw in n° 16 a propos Newton's
binomial series, which is identical to (15), except that, for Newton, the exponent s must be rational.
The formula (15) is thus only Newton's series for s E Z. The real problem,
more difficult, which we shall resolve in Chap. IV, is to prove (15) for all
exponents s E JR. or even C, at least for real z between -1 and 1; the result
which we have just obtained is a small first step in this direction.
Another method for establishing (15) would be to check that on multiplying the series (15) with exponent s by 1 + z one obtains the series for
exponent s + 1. This manifestly reduces to verifying the famous relation
(19.16)
for the binomial coefficients, which, for sEN, explains the no less famous
"Pascal triangle". One lends only to the rich: it was known in the West fully
a century before Pascal, for example to the German Stifel (1486-1567) and
the Italian Tartaglia (1500-1557), and to the Arabs, Indians and Chinese
one or two centuries earlier; but of course they did not write such edifying
Pensees. In fact, the contribution of Pascal - and of Fermat - is to have
exhibited the binomial coefficients in relation to the calculus of combinations
and permutations (whence the birth of the calculus of probabilities) and to
have provided a correct proof, by induction, of the relation (16) for sEN.
To show the usefulness of MacLaurin's formula, let us suppose that the
function sinx is, for x E JR., the sum of a power series in x. It is then easy
to calculate what the series must be by Maclaurin's formula: the successive
derivatives of sin x are cos x, - sin x, - cos x, etc .... and their values at x = °
are 1,0,-1,0,1,0, etc.
II - Convergence: Discrete variables
(19.14)
(1 - z)-P = LP(p + 1) ... (p + n - l)z[n J •
On putting -p = s and replacing z by -z, we get
(19.15)
(1 + z)S = L s(s - 1) ... (s - n + l)z[n J •
If s were a positive integer the coefficient of zn would be zero for n ~ s+ 1 since
it would contain s - s = ° as a factor; the preceding formula would reduce
to the algebraic binomial formula recalled in n° 1. The result obtained shows
that, for s a negative integer, it remains valid on condition:
(i) one writes its coefficients in the form above, and not in the form
s!/n!(s-n)!: this is meaningless apart from the case where sEN and Newton,
who very happily did not know this - it would have led him down a blind alley
-, never did so, contenting himself with the form (15) which has a meaning
for any SEC,
(ii) one replaces the finite sum of algebra by an infinite series,
(iii) one does not forget that the formula obtained is valid only for JzJ < 1,
since otherwise the series is divergent as we saw in n° 16 a propos Newton's
binomial series, which is identical to (15), except that, for Newton, the exponent s must be rational.
The formula (15) is thus only Newton's series for s E Z. The real problem,
more difficult, which we shall resolve in Chap. IV, is to prove (15) for all
exponents s E JR. or even C, at least for real z between -1 and 1; the result
which we have just obtained is a small first step in this direction.
Another method for establishing (15) would be to check that on multiplying the series (15) with exponent s by 1 + z one obtains the series for
exponent s + 1. This manifestly reduces to verifying the famous relation
(19.16)
for the binomial coefficients, which, for sEN, explains the no less famous
"Pascal triangle". One lends only to the rich: it was known in the West fully
a century before Pascal, for example to the German Stifel (1486-1567) and
the Italian Tartaglia (1500-1557), and to the Arabs, Indians and Chinese
one or two centuries earlier; but of course they did not write such edifying
Pensees. In fact, the contribution of Pascal - and of Fermat - is to have
exhibited the binomial coefficients in relation to the calculus of combinations
and permutations (whence the birth of the calculus of probabilities) and to
have provided a correct proof, by induction, of the relation (16) for sEN.
To show the usefulness of MacLaurin's formula, let us suppose that the
function sinx is, for x E JR., the sum of a power series in x. It is then easy
to calculate what the series must be by Maclaurin's formula: the successive
derivatives of sin x are cos x, - sin x, - cos x, etc .... and their values at x = °
are 1,0,-1,0,1,0, etc.