If
K
(
a
n
′
/
1
)
K(a_n’/1)
is a convergent continued fraction with known tails, it can be used to construct modified approximants
f
n
∗
f_n^{\ast }
for other continued fractions
K
(
a
n
/
1
)
K({a_n}/1)
with unknown values. These modified approximants may converge faster to the value
f
f
of
K
(
a
n
/
1
)
K({a_n}/1)
than the ordinary approximants
f
n
{f_n}
do. In particular, if
a
n
−
a
n
′
→
0
{a_n} - a_n’ \to 0
fast enough, we obtain
|
f
−
f
n
∗
|
/
|
f
−
f
n
|
→
0
|f - f_n^{\ast }|/|f - {f_n}| \to 0
; i.e. convergence acceleration. the present paper also gives bounds for this ratio of the two truncation errors, in many cases.