root sequence

x^p=a has a unique root x>0.
There exists an x_1>0 such that x_1^p>a.
Define a sequence x_n>0 by



When p=2,


With a=2 and x_1=3/2, x^2=17/12, and x^3=(17/12+24/17)/2. “This value was once found on a Babylonian tablet of the 18th C BC.”

Graph of y=\frac{x+2/x}{2}, the average of y=x and y=2/x:
root 2 seq


Roger Godement – Analysis I, p108-9

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