Given a sequence
{
x
n
}
n
=
0
∞
\{ {x_n}\} _{n = 0}^\infty
in a Banach space, it is well known that if there is a sequence
{
t
n
}
n
=
0
∞
\{ {t_n}\} _{n = 0}^\infty
such that
‖
x
n
+
1
−
x
n
‖
⩽
t
n
+
1
−
t
n
\left \| {{x_{n + 1}} - {x_n}} \right \| \leqslant {t_{n + 1}} - {t_n}
and
lim
t
n
=
t
∗
>
∞
\lim {t_n} = {t^\ast } > \infty
, then
{
x
n
}
n
=
0
∞
\{ {x_n}\} _{n = 0}^\infty
converges to some
x
∗
{x^\ast }
and the error bounds
‖
x
∗
−
x
n
‖
⩽
t
∗
−
t
n
\left \| {{x^\ast } - {x_n}} \right \| \leqslant {t^\ast } - {t_n}
hold. It is shown that certain stronger hypotheses imply sharper error bounds,
\[
‖
x
∗
−
x
n
‖
⩽
t
∗
−
t
n
(
t
n
−
t
n
−
1
)
μ
‖
x
n
−
x
n
−
1
‖
μ
⩽
t
∗
−
t
n
(
t
1
−
t
0
)
μ
‖
x
1
−
x
0
‖
μ
,
μ
⩾
0.
\left \| {{x^\ast } - {x_n}} \right \| \leqslant \frac {{{t^\ast } - {t_n}}}{{{{({t_n} - {t_{n - 1}})}^\mu }}}{\left \| {{x_n} - {x_{n - 1}}} \right \|^\mu } \leqslant \frac {{{t^\ast } - {t_n}}}{{{{({t_1} - {t_0})}^\mu }}}{\left \| {{x_1} - {x_0}} \right \|^\mu },\quad \mu \geqslant 0.
\]
Representative applications to infinite series and to iterates of types
x
n
=
G
x
n
−
1
{x_n} = G{x_{n - 1}}
and
x
n
=
H
(
x
n
,
x
n
−
1
)
{x_n} = H({x_n},{x_{n - 1}})
are given for
μ
=
1
\mu = 1
. Error estimates with
0
⩽
μ
⩽
2
0 \leqslant \mu \leqslant 2
are shown to be valid and optimal for Newton iterates under the hypotheses of the Kantorovich theorem. The unified convergence theory of Rheinboldt is used to derive error bounds with
0
⩽
μ
⩽
1
0 \leqslant \mu \leqslant 1
for a class of Newton-type methods, and these bounds are shown to be optimal for a subclass of methods. Practical limitations of the error bounds are described.