Newton's iteration is an algorithm for computing the square root of a positive real number via the recurrence equation

(1)

where . For , the recurrence equation converges to with quadratic convergence order. For , every iterate equals 1.

Newton's iteration is simply an application of Newton's method for solving the equation

(2)

For example, when applied numerically, the first few iterations to Pythagoras's constant are 1, 1.5, 1.41667, 1.41422, 1.41421, ....

The first few approximants , , ... to are given by

(3)

For nonnegative integer , these are given by the analytic formula

(4)

For , this also equals , using the principal value of the inverse hyperbolic tangent. The formula can be derived by noting that, for , the recurrence equation can be written as

(5)

which has the closed-form solution

(6)

Solving for then gives the solution derived above.

The following table summarizes the first few iterates for small positive integer

	OEIS	, , ...
1		1, 1, 1, 1, 1, 1, 1, 1, ...
2	A001601/A051009	1, 3/2, 17/12, 577/408, 665857/470832, ...
3	A002812/A071579	1, 2, 7/4, 97/56, 18817/10864, 708158977/408855776, ...