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Prove that it holds for Heron's sequence that xN2 --->2. hint: use 2xn(xn+1 - xn) =...

Prove that it holds for Heron's sequence that xN2 --->2.

hint: use 2xn(xn+1 - xn) = 2 - xN2

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Answer #1

Ans:)

Other form of Heron's sequence can be derived as follows (from the hint):

First we need to prove that the sequence is convergent.

Using AM,GM inequality we can see the following:

Now, divide the heron's sequence derived by x_n,

      

On performing ratio test we get,

(Since , when n tends to infinity)

Suppose that as n tends to infinity, the sequence tends to some limiting value l such that

The equation becomes,

Thus

Hence proved.

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