§2. Absolutely convergent series
105
so that, arguing like Euler, one might deduce that the total of the nth sum
also tends to o. But it is equal to 1 for any n.
Later we shall meet other cases where Euler passed to the limit automatically in a sequence of convergent series as if "it went without saying".
Now it is quite difficult to believe that he did not know the particularly
trivial counterexample which we have just presented, and which spices up
the "marvels of the harmonic series". Conclusion? The most charitable is
that he employed numerical methods to confirm his intuitions 28 and that,
on the contrary, not having available the methods invented in the XIX th
century to justify his calculations, he preferred (rightly) to advance rather
than to abstain from publishing such beautiful (and correct) formulae. All
his contemporaries were content to admire them, without asking questions
of themselves or of him: if "Euler says that ... ", it is surely true. One meets
the same reaction nowadays, no longer a propos theorems or formulae, but
conjectures.
We can now justify what we have said in nO 5, example 7, a propos logarithms:
Theorem 3. For all x> 0, the sequence with general term n(xl/n -1) tends
to a limit f (x). The function f is Coo, strictly increasing, and satisfies
(10.7)
f(xy) = f(x) + f(y),
J'(x) = l/x.
Suppose first that x > 1 and put
(10.8)
un = n(xl/n - 1),
whence Un> o. One has
(10.9)
x = (1 + un/n)n for all n.
Now we saw in Example 2 that, for all U > 0, the sequence (1 + u/n)n is
increasing. For u = Un one thus has
x = (1 + un/nt < [1 + un/(n + l)t+1 ,
whence, using (9) for nand n + 1,
28 See in Hairer and Wanner, Analysis by Its History (Springer, 1995), p. 25, a table
of values of (1 + l/n)n and 1 + 1/1! + ... + l/n! for n ::; 28. The second sequence
provides, for n = 28, the 29 first decimals of the limit e = exp 1 = 2.718 ... ,
though the first, for n = 28, is equal to 2.671. Recourse to brute numerical
calculation to verify formula (14) thus presupposes rather a lot of work. A more
practical method, at that time, would have been to use a table of logarithms -
they had been available to 20 decimals for some time. Now log [(1 + x/n)n] =
n.log(1 + x/n), so that, for x < 1 for example, it would be possible to calculate
without difficulty the left hand side exact to 10 decimal places for n = 10 6 and
to deduce a much better approximation to (1 + x/n)n than by brute calculation.
105
so that, arguing like Euler, one might deduce that the total of the nth sum
also tends to o. But it is equal to 1 for any n.
Later we shall meet other cases where Euler passed to the limit automatically in a sequence of convergent series as if "it went without saying".
Now it is quite difficult to believe that he did not know the particularly
trivial counterexample which we have just presented, and which spices up
the "marvels of the harmonic series". Conclusion? The most charitable is
that he employed numerical methods to confirm his intuitions 28 and that,
on the contrary, not having available the methods invented in the XIX th
century to justify his calculations, he preferred (rightly) to advance rather
than to abstain from publishing such beautiful (and correct) formulae. All
his contemporaries were content to admire them, without asking questions
of themselves or of him: if "Euler says that ... ", it is surely true. One meets
the same reaction nowadays, no longer a propos theorems or formulae, but
conjectures.
We can now justify what we have said in nO 5, example 7, a propos logarithms:
Theorem 3. For all x> 0, the sequence with general term n(xl/n -1) tends
to a limit f (x). The function f is Coo, strictly increasing, and satisfies
(10.7)
f(xy) = f(x) + f(y),
J'(x) = l/x.
Suppose first that x > 1 and put
(10.8)
un = n(xl/n - 1),
whence Un> o. One has
(10.9)
x = (1 + un/n)n for all n.
Now we saw in Example 2 that, for all U > 0, the sequence (1 + u/n)n is
increasing. For u = Un one thus has
x = (1 + un/nt < [1 + un/(n + l)t+1 ,
whence, using (9) for nand n + 1,
28 See in Hairer and Wanner, Analysis by Its History (Springer, 1995), p. 25, a table
of values of (1 + l/n)n and 1 + 1/1! + ... + l/n! for n ::; 28. The second sequence
provides, for n = 28, the 29 first decimals of the limit e = exp 1 = 2.718 ... ,
though the first, for n = 28, is equal to 2.671. Recourse to brute numerical
calculation to verify formula (14) thus presupposes rather a lot of work. A more
practical method, at that time, would have been to use a table of logarithms -
they had been available to 20 decimals for some time. Now log [(1 + x/n)n] =
n.log(1 + x/n), so that, for x < 1 for example, it would be possible to calculate
without difficulty the left hand side exact to 10 decimal places for n = 10 6 and
to deduce a much better approximation to (1 + x/n)n than by brute calculation.