§3. Bolzano-Weierstrass and Cauchy's criterion
235
Theorem 14 (Cauchy). The series L Un converges if and only if for every
number r > 0 there exists an integer N such that
(11.1 )
Iup + ... + uql < r for N < p::::; q.
Since the left hand side of (1) is smaller than the analogous expression
for the series with positive terms L I Un I, convergence of the latter will all the
more imply (1), and so convergence of the initial series. So we again find,
with Cauchy, the fact that every absolutely convergent series converges, but
with quite another proof.
Cauchy's criterion also allows one to establish more subtle results than
Leibniz' Theorem for alternating series.
Theorem 15 (Dirichlet). Let LUn be a series whose partial sums are
bounded and let (vn ) be a sequence of positive numbers which decreases to o.
Then the series L Un Vn converges.
Put
for p < q (put Sp-l = 0 if P = 1): then
upvp + ... + UqVq = (sp - Sp-l)Vp + ... + (Sq - Sq-l)Vq =
= -Sp-1Vp + sp(vp - vp+d + ... + Sq-l(Vq-l - vq) + SqVq.
Since Vn 2 Vn+l 20, it follows that
IUpvp+ ... +uqvql ::::; Isp-dvp+ Ispl(vp -Vp+1) + ... + ISq-ll(vq-l -vq) + ISqlvq.
By hypothesis, sup ISnl = M < +00, whence
(11.2)
Iupvp + ... + uqvql ::::;
::::; M [vp + (vp - Vp+l) + ... + (Vq-l - vq) + vq] = 2Mvp;
since vp tends to 0 the left hand side is < E: for p large, whence Cauchy's
criterion for L UnVn' qed.
In 1826 Abel established just the inequality (2) for p = 1 without assuming
that the decreasing sequence Vn tends to 0 nor drawing a conclusion as to
the convergence of the series, yet this is clearly the essential point. In this
more general case one again obtains the conclusion of Dirichlet's Theorem
on strengthening the hypothesis imposed on the series Un. Now Vn certainly
tends to a limit v 2 0 so that, if the partial sums of the series L Un are
bounded, the series L Un (vn - v) will converge, by Theorem 15. To deduce
the convergence of L Un Vn it then suffices to suppose the series Un convergent.
Finally:
Theorem 15' (Abel). Let LUn be a convergent series and (vn ) a decreasing sequence of positive numbers. Then the series L Un Vn converges.
235
Theorem 14 (Cauchy). The series L Un converges if and only if for every
number r > 0 there exists an integer N such that
(11.1 )
Iup + ... + uql < r for N < p::::; q.
Since the left hand side of (1) is smaller than the analogous expression
for the series with positive terms L I Un I, convergence of the latter will all the
more imply (1), and so convergence of the initial series. So we again find,
with Cauchy, the fact that every absolutely convergent series converges, but
with quite another proof.
Cauchy's criterion also allows one to establish more subtle results than
Leibniz' Theorem for alternating series.
Theorem 15 (Dirichlet). Let LUn be a series whose partial sums are
bounded and let (vn ) be a sequence of positive numbers which decreases to o.
Then the series L Un Vn converges.
Put
for p < q (put Sp-l = 0 if P = 1): then
upvp + ... + UqVq = (sp - Sp-l)Vp + ... + (Sq - Sq-l)Vq =
= -Sp-1Vp + sp(vp - vp+d + ... + Sq-l(Vq-l - vq) + SqVq.
Since Vn 2 Vn+l 20, it follows that
IUpvp+ ... +uqvql ::::; Isp-dvp+ Ispl(vp -Vp+1) + ... + ISq-ll(vq-l -vq) + ISqlvq.
By hypothesis, sup ISnl = M < +00, whence
(11.2)
Iupvp + ... + uqvql ::::;
::::; M [vp + (vp - Vp+l) + ... + (Vq-l - vq) + vq] = 2Mvp;
since vp tends to 0 the left hand side is < E: for p large, whence Cauchy's
criterion for L UnVn' qed.
In 1826 Abel established just the inequality (2) for p = 1 without assuming
that the decreasing sequence Vn tends to 0 nor drawing a conclusion as to
the convergence of the series, yet this is clearly the essential point. In this
more general case one again obtains the conclusion of Dirichlet's Theorem
on strengthening the hypothesis imposed on the series Un. Now Vn certainly
tends to a limit v 2 0 so that, if the partial sums of the series L Un are
bounded, the series L Un (vn - v) will converge, by Theorem 15. To deduce
the convergence of L Un Vn it then suffices to suppose the series Un convergent.
Finally:
Theorem 15' (Abel). Let LUn be a convergent series and (vn ) a decreasing sequence of positive numbers. Then the series L Un Vn converges.