and
,
and, for 
, define
Then, .gif)
and
.gif)
converge
(quadratically) to the common limit
.
Notice that
is the
arithmetic mean
(average) of 
and 
, while
is the
harmonic mean
of and
. [To
see that this is truly a generalization of the Babylonian
algorithm, set =
/.]
To see that
the algorithm converges, first convince yourself, using induction,
that
for 
> 1. That is, the
's
decrease while
the 's
increase. Thus, both
sequences converge. The only issue here is finding the limit.
Next, we use
nearly the same computation as for our first algorithm:
As before, and
are half
as far apart as
and .
By induction,
and hence
and
have the same limit
.
To compute this limit, notice that
and hence 
;
that is, 
and 
.
As
with the Babylonian algorithm, now that we know the limit we can
discuss the rate of convergence. In this setting we have
Thus, the sequence
converges
quadratically. A similar computation shows that
is likewise
quadratically convergent.
Iterative
algorithms of this type are actually quite common and,
in recent years, have been used to good advantage in constructing
fast, effective algorithms for computing various transcendental
functions and, in particular, for computing
to millions of
places. Another such algorithm is
described in connection with our discussion of
Archimedes' method.