Now we claim
that
,
we begin by making a first "guess" 
(it doesn't need to be a "good" guess, even
= 1 will do). Then, as we've
seen, the number
/
is also an estimate for .
Now we claim
that
the average of our first two estimates, is an even better estimate
for .
Assuming this for the moment then means that
should improve upon both of our previous estimates.
But why
stop there? Continuing, we define
As 
increases, each
successive estimate 
should be ever
closer to .
In fact, we have
provided that we begin with 
(and > 0).
Before
we check this claim,
let's look at a few examples.