268
III - Convergence: Continuous variables
for all x E X = K - {a}. On taking the least upper bound of the left hand
side for x EX, we find
and since the f~ converge uniformly on K by hypothesis, the Un satisfy
Cauchy's criterion for uniform convergence on X. Whence hypothesis (ii) of
Theorem 16, qed.
Corollary. Let E fn(x) be a series of functions defined and differentiable on
an interval I. Suppose that the series converges at a point of I and that the
derived series converges normally on every compact K c I. Then the given
series converges normally on every K c I, its sum s(x) is differentiable, and
s'(x) = Ef~(x).
We need only apply the theorem to the partial sums sn(x) of the given
series, and observe that normal convergence of the derived series implies
uniform convergence of the sequence s~(x) on every compact set. Normal
convergence of the given series follows from inequality (2).
Example 1. Let f(x) = Eanx n be a power series converging for Ixl < R,
R > 0, and let us work on I =]- R, R[. We know (Chap. II, nO 19) that the
series E nanx n - 1 of derivatives also converges for Ixl < R and so converges
normally on all compact intervals K = [-r,r] with r < R. Theorem 19 then
applies, confirming that one can differentiate a power series term-by-term, at
least in the real domain.
Example 2. Consider a Fourier series
f(x) = L an cos nx + bn sin nx
and suppose that the series E nan and E nbn converge absolutely (examples:
an = link with k > 2, or an = qn with Iql < 1, etc.) Since sine and cosine
are ~ 1 in modulus, the series obtained by differentiating the given series
term-by-term will converge normally, whence
f'(x) = L(nbn cosnx - nan sin nx).
The square wave series clearly does not fit into this framework. If you consider
the series f(x) = Esin(n 2 x)ln 2 at which Riemann, Weierstrass, and surely
many others, tilted without success up to 1930, the argument fails totally,
and must do so, since f(x) has a strong tendency not to be differentiable
except at very exceptional points.
Example 3. Consider the series «s) = L: 11n s for S > 1. We shall see in
Chap. IV, where we shall define real powers - they behave exactly like integral
III - Convergence: Continuous variables
for all x E X = K - {a}. On taking the least upper bound of the left hand
side for x EX, we find
and since the f~ converge uniformly on K by hypothesis, the Un satisfy
Cauchy's criterion for uniform convergence on X. Whence hypothesis (ii) of
Theorem 16, qed.
Corollary. Let E fn(x) be a series of functions defined and differentiable on
an interval I. Suppose that the series converges at a point of I and that the
derived series converges normally on every compact K c I. Then the given
series converges normally on every K c I, its sum s(x) is differentiable, and
s'(x) = Ef~(x).
We need only apply the theorem to the partial sums sn(x) of the given
series, and observe that normal convergence of the derived series implies
uniform convergence of the sequence s~(x) on every compact set. Normal
convergence of the given series follows from inequality (2).
Example 1. Let f(x) = Eanx n be a power series converging for Ixl < R,
R > 0, and let us work on I =]- R, R[. We know (Chap. II, nO 19) that the
series E nanx n - 1 of derivatives also converges for Ixl < R and so converges
normally on all compact intervals K = [-r,r] with r < R. Theorem 19 then
applies, confirming that one can differentiate a power series term-by-term, at
least in the real domain.
Example 2. Consider a Fourier series
f(x) = L an cos nx + bn sin nx
and suppose that the series E nan and E nbn converge absolutely (examples:
an = link with k > 2, or an = qn with Iql < 1, etc.) Since sine and cosine
are ~ 1 in modulus, the series obtained by differentiating the given series
term-by-term will converge normally, whence
f'(x) = L(nbn cosnx - nan sin nx).
The square wave series clearly does not fit into this framework. If you consider
the series f(x) = Esin(n 2 x)ln 2 at which Riemann, Weierstrass, and surely
many others, tilted without success up to 1930, the argument fails totally,
and must do so, since f(x) has a strong tendency not to be differentiable
except at very exceptional points.
Example 3. Consider the series «s) = L: 11n s for S > 1. We shall see in
Chap. IV, where we shall define real powers - they behave exactly like integral