§2. Series expansions
361
Isp - sl < L lup(n) - u(n)1 = L lup(n) - u(n)1 + L lup(n) - u(n)1 ~
n
n>N
< L lup(n) - u(n)1 + 2 L v(n).
n~N
n>N
Take an r > o. The second sum, which does not depend on p, is < r for all
sufficiently large N since the series E v( n) converges. Choosing such an N,
the first sum tends to 0 as p - t 00 since so do the N differences which
compose it; it is therefore < r once p exceeds a suitably chosen integer PN.
Then Isp - sl < 3r for P sufficiently large, qed.
We can now state the result formally:
Theorem 10. We have
(12.8)
for -1 < x ~ 1.
Newton and Mercator, who were the first to obtain this result, in about
1665-1668, proceeded a little differently: it was already known that log( 1 + x)
is the integral from 0 to x of the function 1 1(1 + x) = 1 - x + x 2 - .•• ; since
the integral of a sum (though it be infinite ... ) offunctions is the sum of the
integrals of the latter, and since it was already known how to integrate xm
for m an integer, thanks to Fermat, Cavalieri and Wallis, the series appeared
of its own accord.
The crucial point here, naturally, is the connection between logarithms
and the area of the hyperbola. It would have been obvious to Napier himself
if he had known the "fundamental theorem of the differential and integral
calculus", which enables one to calculate the area bounded by a curve y =
f(x) if one knows an F for which f(x) = F'(x). Let us explain the history of
the subject.
Following the inventor of logarithms, one considers a point y which moves
from 20 10 7 to 0 at a speed inversely proportional to the time x, in other words
satisfying the relation y'(x) = -10 7 Ix; the connection with the area of the
hyperbola is obvious to us now, but not, of course, to his contemporaries and
immediate successors. Furthermore, his tables, and those of Briggs to base
10, were not considered as mathematical theories: they were aids to numerical
calculation, intended, doubtless, mainly for the astronomers in Napier's case
(he wrote in Latin); according to Edward Wright, who translated them into
English with a dedication to the East India Company and to Briggs, who
completed them and transformed them into log to base 10, they are above
all for the use of navigators, who worried little about the Latin but adopted
the log with enthusiasm.
20 Napier constructed a table of the function log cos x, and since the use of decimal
fractions was only in its infancy, he tabulated -10 7 log cos x in order to obtain
positive integral values for log.
361
Isp - sl < L lup(n) - u(n)1 = L lup(n) - u(n)1 + L lup(n) - u(n)1 ~
n
n>N
< L lup(n) - u(n)1 + 2 L v(n).
n~N
n>N
Take an r > o. The second sum, which does not depend on p, is < r for all
sufficiently large N since the series E v( n) converges. Choosing such an N,
the first sum tends to 0 as p - t 00 since so do the N differences which
compose it; it is therefore < r once p exceeds a suitably chosen integer PN.
Then Isp - sl < 3r for P sufficiently large, qed.
We can now state the result formally:
Theorem 10. We have
(12.8)
for -1 < x ~ 1.
Newton and Mercator, who were the first to obtain this result, in about
1665-1668, proceeded a little differently: it was already known that log( 1 + x)
is the integral from 0 to x of the function 1 1(1 + x) = 1 - x + x 2 - .•• ; since
the integral of a sum (though it be infinite ... ) offunctions is the sum of the
integrals of the latter, and since it was already known how to integrate xm
for m an integer, thanks to Fermat, Cavalieri and Wallis, the series appeared
of its own accord.
The crucial point here, naturally, is the connection between logarithms
and the area of the hyperbola. It would have been obvious to Napier himself
if he had known the "fundamental theorem of the differential and integral
calculus", which enables one to calculate the area bounded by a curve y =
f(x) if one knows an F for which f(x) = F'(x). Let us explain the history of
the subject.
Following the inventor of logarithms, one considers a point y which moves
from 20 10 7 to 0 at a speed inversely proportional to the time x, in other words
satisfying the relation y'(x) = -10 7 Ix; the connection with the area of the
hyperbola is obvious to us now, but not, of course, to his contemporaries and
immediate successors. Furthermore, his tables, and those of Briggs to base
10, were not considered as mathematical theories: they were aids to numerical
calculation, intended, doubtless, mainly for the astronomers in Napier's case
(he wrote in Latin); according to Edward Wright, who translated them into
English with a dedication to the East India Company and to Briggs, who
completed them and transformed them into log to base 10, they are above
all for the use of navigators, who worried little about the Latin but adopted
the log with enthusiasm.
20 Napier constructed a table of the function log cos x, and since the use of decimal
fractions was only in its infancy, he tabulated -10 7 log cos x in order to obtain
positive integral values for log.