§4. Differentiable functions
269
powers -, that the function 8 ~ n S = exp(8.logn) is differentiable and has
derivative n S log n. The series of derivatives
is, like the Riemann series itself (nO 8, example 2), normally convergent on
[0", +oo[ for any 0" > 1. We shall show later (Chap. IV, nO 5) that for any real
number k > 0 we have
log x = o(xk) when x -+ +00;
up to a constant factor one has (logn)/n s :::; l/n s - k :::; l/n u - k for n large; by
choosing k > 0 small enough that 0" - k > 1, i.e. 0 < k < 0" - 1, one obtains
normal convergence. So we can differentiate term-by-term, whence
('(8) = - L)logn)/n s
for 8 > 1, and we can repeat this operation ad libitum.
Example 4. In Chap. II, nO 21, we asked the question of whether, from the
relation
(17.5)
7r cot 7rX = l/x + L x/n(x - n)
where one sums over all nonzero nEil, one can, by differentiation, deduce
~he formula
(17.6)
where one sums over all the nEil without exception. For this, let us work
on a compact interval of the form K = [-p, p] with pEN and split the
series in (5) as the sum of terms for which Inl :::; p and the sum of the other
terms. The first has only a finite number of terms, so can be differentiated
term-by-term, and one thus obtains, in (6), the corresponding partial sum.
AP, for the sum of the terms for which the index satisfies Inl > p, it is, in (5), a
series of functions defined and differentiable on all of K and which converges
everywhere on K. For Theorem 19 to be applicable it is thus enough to show
that the series L(x - n)-2, taken over those n such that Inl > p, converges
normally on K.
Now the equality n = (n - x) + x shows that
Inl ::s Ix - nl + Ixl ::s Ix - nl + p,
Whence Ix - nl ~ Inl - p > 0 and thus
1/{x - n)2 ::s 1/{lnl - p)2 = v{n).