176
II - Convergence: Discrete variables
a polynomial in X and X-I ... One then extends this expression to the general case, remaining aware of the fact that this is a pure convention to ease
the calculations - one really can calculate with these "series" as if they were
polynomials in X and X-I - and not a theorem in the genre "limit of partial
sums", an expression with no meaning in algebra. The first mathematician
deliberately to use formal series (with coefficients in q was, naturally, Euler
who, of course, provided neither justifications nor explanations: he calculated.
One can consider a power series [or even a series with possibly a finite
number of terms of negative degree - cf. cot xl as a formal series with coefficients in C, but the difference between algebra and analysis lies in the fact
that the formal series which converge only for x = 0, for example L n!x n ,
have no interest in analysis as we have already noted in nO 15, example 4.
The miracle lies in the fact that all the reasonable operations - and not only
the algebraic, consider derivation for a start - that one performs, in C, on
convergent formal series, those having a radius of convergence> 0, lead again
to convergent series.
All the same one must be prudent. If for example you calculate the expression 1/ cos z a la Newton, replacing cos z by its power series, you will
again find a convergent power series by Theorem 17. But while that for cos z
converges for any z, the new series will converge in no more than (and, in
fact, exactly in) the largest disc with centre 0 not containing any zero of the
function cos z, in other words for Izl < 7r /2. Such a result could never be
established if one were restricted to examining the coefficients of this series:
they are quite complicated and 7r does not appear in them; one needs the
general theory of analytic functions: if a function f is analytic in an open
subset G of C, then for all a E G the Taylor series of f about a converges
and represents f on the largest open disc with centre a contained in G; we
will show this in Chap. VII. For the function 1/ cosz, G is the set of z E C
where cos z -I- 0, whence one finds the radius of convergence, so long as one
knows that, in C as in JR, the series vanishes only at the points z = (2k+ 1)7r /2.
Newton's method, in its simplest form, allows one to calculate the coefficients in the formula (21.4), i.e. to justify the series (21.5) for the function
cot x, subject to the assumption of the existence of such a formula - this detail would have held back neither Newton nor his successors for 150 years -
and to accepting the series expansions of the functions sine and cosine already
mentioned.
The first point raises no difficulty. Indeed
cot x = cos x/ sin x = (1 - x 2 /2! + .. . )/x(1 - x 2 /3! + ... ),
so that x. cot x is the quotient of two power series starting with 1. Theorem 17
was introduced precisely to show that this quotient is itself a power series
II - Convergence: Discrete variables
a polynomial in X and X-I ... One then extends this expression to the general case, remaining aware of the fact that this is a pure convention to ease
the calculations - one really can calculate with these "series" as if they were
polynomials in X and X-I - and not a theorem in the genre "limit of partial
sums", an expression with no meaning in algebra. The first mathematician
deliberately to use formal series (with coefficients in q was, naturally, Euler
who, of course, provided neither justifications nor explanations: he calculated.
One can consider a power series [or even a series with possibly a finite
number of terms of negative degree - cf. cot xl as a formal series with coefficients in C, but the difference between algebra and analysis lies in the fact
that the formal series which converge only for x = 0, for example L n!x n ,
have no interest in analysis as we have already noted in nO 15, example 4.
The miracle lies in the fact that all the reasonable operations - and not only
the algebraic, consider derivation for a start - that one performs, in C, on
convergent formal series, those having a radius of convergence> 0, lead again
to convergent series.
All the same one must be prudent. If for example you calculate the expression 1/ cos z a la Newton, replacing cos z by its power series, you will
again find a convergent power series by Theorem 17. But while that for cos z
converges for any z, the new series will converge in no more than (and, in
fact, exactly in) the largest disc with centre 0 not containing any zero of the
function cos z, in other words for Izl < 7r /2. Such a result could never be
established if one were restricted to examining the coefficients of this series:
they are quite complicated and 7r does not appear in them; one needs the
general theory of analytic functions: if a function f is analytic in an open
subset G of C, then for all a E G the Taylor series of f about a converges
and represents f on the largest open disc with centre a contained in G; we
will show this in Chap. VII. For the function 1/ cosz, G is the set of z E C
where cos z -I- 0, whence one finds the radius of convergence, so long as one
knows that, in C as in JR, the series vanishes only at the points z = (2k+ 1)7r /2.
Newton's method, in its simplest form, allows one to calculate the coefficients in the formula (21.4), i.e. to justify the series (21.5) for the function
cot x, subject to the assumption of the existence of such a formula - this detail would have held back neither Newton nor his successors for 150 years -
and to accepting the series expansions of the functions sine and cosine already
mentioned.
The first point raises no difficulty. Indeed
cot x = cos x/ sin x = (1 - x 2 /2! + .. . )/x(1 - x 2 /3! + ... ),
so that x. cot x is the quotient of two power series starting with 1. Theorem 17
was introduced precisely to show that this quotient is itself a power series