386
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
expx = lim(l + x/n)n
is witness to this. On pushing the false "argument" to the limit without
having the same intuition as Euler for correct formulae, one might even claim
that 1 + x/n = 1 for n infinitely large, and so that (1 + x/n)n = In = 1
- one sometimes meets this in students' work -, which would yield the relation
exp(x) = 1 for any x.
Its social impact would be immense as one sees from an argument due
to Jakob Bernoulli, 1690, in an age when financial mathematics - interest,
insurance, games of chance, etc. - interested many other people, including
de Moivre, Leibniz and the English, waiting for Euler himself and his "tontines" .
Suppose that you deposit your fortune for a year with a banker who,
rather than calculating at the end of the year only the interest that he owes
you the rate of x% per annum, instead allows at each instant for what he owes
at the preceding instant. If, a little less generously, he does this n times each
year, your balance would be multiplied by 1 + x/n after the first period of
365/n days, then by (1 +x/n)2 after the second and so on until your fortune,
at the end of the year, is multiplied by (1 + x/n)n. This has been known for
as long as moneylenders have existed - one in finds it already appearing in
Babylonians examples! If you pass to the limit you will find that it has been
multiplied by the factor exp(x) - or by 1 if you believe that lim(l+x/n)n = l.
This fantasy formula allows you to prove that if you deposited your fortune with an infinitely honest banker, it would not increase even by a penny
at the end of the year, and this at any rate of interest generously offered by
the philanthropist. This would be the Triumph of Virtue: total honesty would
cost strictly nothing to those who practiced it, and would bring strictly nothing to those who benefit from it. The same argument would show conversely
that, if your banker lends you money at 1000% per annum and calculates at
each instant the compound interest that you owe him, you will have to give
him strictly no more, at the end of the year, than the sum he gave you at the
beginning; this time, the Bankruptcy of the Sharks. One sees revolution.
In fact, the discoverer of the power series of cos x and sin x, namely Newton, proceeded in quite another way: as we saw in nO 14, formula (24), an
integration gave him directly, via his binomial formula, the power series for
arcsinx and it was by inverting it that Newton found that of sinx. The
problem is to find a power series
(15.5)
satisfying the relation
(15.6)
x = y + y3/3.2 + l.3y5/5.2.4 +... (= arcsiny),
IV - Powers, Exponentials, Logarithms, Trigonometric Functions
expx = lim(l + x/n)n
is witness to this. On pushing the false "argument" to the limit without
having the same intuition as Euler for correct formulae, one might even claim
that 1 + x/n = 1 for n infinitely large, and so that (1 + x/n)n = In = 1
- one sometimes meets this in students' work -, which would yield the relation
exp(x) = 1 for any x.
Its social impact would be immense as one sees from an argument due
to Jakob Bernoulli, 1690, in an age when financial mathematics - interest,
insurance, games of chance, etc. - interested many other people, including
de Moivre, Leibniz and the English, waiting for Euler himself and his "tontines" .
Suppose that you deposit your fortune for a year with a banker who,
rather than calculating at the end of the year only the interest that he owes
you the rate of x% per annum, instead allows at each instant for what he owes
at the preceding instant. If, a little less generously, he does this n times each
year, your balance would be multiplied by 1 + x/n after the first period of
365/n days, then by (1 +x/n)2 after the second and so on until your fortune,
at the end of the year, is multiplied by (1 + x/n)n. This has been known for
as long as moneylenders have existed - one in finds it already appearing in
Babylonians examples! If you pass to the limit you will find that it has been
multiplied by the factor exp(x) - or by 1 if you believe that lim(l+x/n)n = l.
This fantasy formula allows you to prove that if you deposited your fortune with an infinitely honest banker, it would not increase even by a penny
at the end of the year, and this at any rate of interest generously offered by
the philanthropist. This would be the Triumph of Virtue: total honesty would
cost strictly nothing to those who practiced it, and would bring strictly nothing to those who benefit from it. The same argument would show conversely
that, if your banker lends you money at 1000% per annum and calculates at
each instant the compound interest that you owe him, you will have to give
him strictly no more, at the end of the year, than the sum he gave you at the
beginning; this time, the Bankruptcy of the Sharks. One sees revolution.
In fact, the discoverer of the power series of cos x and sin x, namely Newton, proceeded in quite another way: as we saw in nO 14, formula (24), an
integration gave him directly, via his binomial formula, the power series for
arcsinx and it was by inverting it that Newton found that of sinx. The
problem is to find a power series
(15.5)
satisfying the relation
(15.6)
x = y + y3/3.2 + l.3y5/5.2.4 +... (= arcsiny),