§2. Absolutely convergent series
121
shows that also
(13.4)
If one considers for example the series
1 - 1/3 + 1/5 - 1/7 ... ,
for which Un = 1/(2n -1), whence Ur+l - Ur+2 = 2/(2r + 1)(2r + 3), one has
2/(2r + 1)(2r + 3) :::; Is - sri:::; 1/(2r + 1).
In order to calculate s exact to 20 decimal places, one must therefore calculate
Sr to the same precision for an integer r such that
2/(2r + 1)(2r + 3) < 10- 20 ,
i.e. such that (2r + 1)(2r + 3) > 2.10 20 , which requires (2r + 3)2 > 2.10 20 , so
2r + 3> 1.4 X 1010; one must therefore calculate about seven billion terms of
the series and their sum, all exact to 20 decimal places. Since errors have a
strong tendency to reinforce, and since 10- 20 is almost equal to 7.10 9 .10- 31 ,
the calculations must be performed to 31 exact decimals. Exercise: calculate 1/1234567898 by hand to 31 exact decimals (in fact 21, since the result
obviously starts with ten Os).
One will therefore understand the hilarity - a very rare event, it seems
- which must have seized Newton when in 1676, on the occasion of a brief
exchange of correspondence with Leibniz (1646-1716) - an exchange in the
course of which each very obviously strained to show that he knew more than
the other, particularly Newton who was several years in advance but had published nothing -, Leibniz informed him of a sort of mechanical procedure for
calculating an arbitrarily high number of decimals of IT, namely the formula
(not obvious at this stage of the exposition)
IT /4 = 1 - 1/3 + 1/5 - 1/7 + ....
Newton who a dozen years earlier, had amused himself - one amuses oneself
as one can, and he adduced his youth in excusing himself to Leibniz - in
calculating log(1 + x) to about 50 places, with the aid of the analogous series
x - x 2 /2 + x 3 /3 - ... , but only for values of x very close to 0 and not,
for example, for x = 1, a desperate case, replied to Leibniz that it would
require a million years of work to calculate the first twenty decimals of IT
by his method, and further that his formula was already known in 1671 to
his compatriot James Gregory. One notes, a curious coincidence, that in 1673
Leibniz had presented his calculating machine to the Royal Society of London
and to the Academie des sciences of Paris. It was inspired by that of Pascal
and capable of performing not only additions and subtractions like that one,
but also multiplications and divisions; Leibniz developed it over decades, at
121
shows that also
(13.4)
If one considers for example the series
1 - 1/3 + 1/5 - 1/7 ... ,
for which Un = 1/(2n -1), whence Ur+l - Ur+2 = 2/(2r + 1)(2r + 3), one has
2/(2r + 1)(2r + 3) :::; Is - sri:::; 1/(2r + 1).
In order to calculate s exact to 20 decimal places, one must therefore calculate
Sr to the same precision for an integer r such that
2/(2r + 1)(2r + 3) < 10- 20 ,
i.e. such that (2r + 1)(2r + 3) > 2.10 20 , which requires (2r + 3)2 > 2.10 20 , so
2r + 3> 1.4 X 1010; one must therefore calculate about seven billion terms of
the series and their sum, all exact to 20 decimal places. Since errors have a
strong tendency to reinforce, and since 10- 20 is almost equal to 7.10 9 .10- 31 ,
the calculations must be performed to 31 exact decimals. Exercise: calculate 1/1234567898 by hand to 31 exact decimals (in fact 21, since the result
obviously starts with ten Os).
One will therefore understand the hilarity - a very rare event, it seems
- which must have seized Newton when in 1676, on the occasion of a brief
exchange of correspondence with Leibniz (1646-1716) - an exchange in the
course of which each very obviously strained to show that he knew more than
the other, particularly Newton who was several years in advance but had published nothing -, Leibniz informed him of a sort of mechanical procedure for
calculating an arbitrarily high number of decimals of IT, namely the formula
(not obvious at this stage of the exposition)
IT /4 = 1 - 1/3 + 1/5 - 1/7 + ....
Newton who a dozen years earlier, had amused himself - one amuses oneself
as one can, and he adduced his youth in excusing himself to Leibniz - in
calculating log(1 + x) to about 50 places, with the aid of the analogous series
x - x 2 /2 + x 3 /3 - ... , but only for values of x very close to 0 and not,
for example, for x = 1, a desperate case, replied to Leibniz that it would
require a million years of work to calculate the first twenty decimals of IT
by his method, and further that his formula was already known in 1671 to
his compatriot James Gregory. One notes, a curious coincidence, that in 1673
Leibniz had presented his calculating machine to the Royal Society of London
and to the Academie des sciences of Paris. It was inspired by that of Pascal
and capable of performing not only additions and subtractions like that one,
but also multiplications and divisions; Leibniz developed it over decades, at