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II - Convergence: Discrete variables
The convergence of a sequence (un) of complex numbers being equivalent
to that of the series E(un - Un+l), any convergence criterion applicable to
series will provide a criterion for sequences. The most obvious follows from
Theorem 4 for series with positive terms:
Theorem 11. For a sequence (un) of complex numbers to converge it suffices
that E IUn - un+ll < +00.
This theorem applies principally in the case where one has an estimate of
the form IUn - un+ll < Mqn with 0 < q < 1, or else < Mink with k > 1,
where M is a positive constant.
As an illustration, consider an interval I c JR, a map f : I ---> I, and let
us set ourselves to solve the equation
(16.9)
f(x) = x
in I. The method of iteration consists of choosing an Xo E I arbitrarily, and
then examining the points
(16.10)
Xl = f(xo),
The figures below demonstrate what can happen.
fig. 5. Walter, Analysis 1, p. 314