TITLE: How is the sequence 1, 1.4, 1.41, 1.414 generated?
QUESTION [13 upvotes]: In many of the standard textbooks discussing Real Numbers, the Cauchy sequence that converges to $\sqrt{2}$ is given as
1, 1.4, 1.41, 1.414, 1.4142, ...
or
2, 1.5, 1.42, 1.415, 1.4143, ...
My question is how are these sequences generated? In other words, if I have 1, 1.4, 1.41 how do I figure out that the next element of the sequence is 1.414?

REPLY [2 votes]: Other answers give efficient ways of finding additional terms of the Cauchy sequence, or sequences that may be better than the Cauchy series for practical purposes.
But as far as I'm concerned, the Cauchy sequence has nothing to do with practical computations. It is simply either a way of defining the real numbers or (if you prefer Dedekind cuts as a definition) a way of justifying infinite decimal representations of real numbers.
The Cauchy sequences $1, 1.4, 1.41, 1.414, 1.4142,\ldots$ and
$2, 1.5, 1.42, 1.415, 1.4143, \ldots$ are well defined because after each step you have narrowed down the possible values of $\sqrt2$ to an interval of width $10^{-n}$ for some integer $n,$ and for the next step you say which of the $10$ equal subdivisions of that interval contains $\sqrt2.$
Since the point here is simply that there is an answer, not that we know how to find it quickly, you can appeal to "brute force."
For example, after the upper and lower Cauchy sequences around $\sqrt2$
reach the terms $1.414$ and $1.415,$ respectively, the next term in each sequence has to come from the following set:
$$\{1.4140, 1.4141, 1.4142, 1.4143, 1.4144, 1.4145, 1.4146, 1.4147, 1.4148, 1.4149, 1.4150\}.$$
It takes only a finite number of steps to square all of these values and find out which ones are less than $\sqrt2$ and which are greater.
(You already know two of the squares, so you only have to compute the other nine.)
Once you have done this, you find that
$1.4142^2 < 2 < 1.4143^2,$ so $\sqrt2$ is between 
$1.4142$ and $1.4143,$ which are therefore the next terms in the lower and upper sequences, respectively.
All the rest is optimization of the computations.