'Suppose that A is any positive real number: Define the sequence (an} recursively by an+i 2 with n € N and with 01 an arbitrary positive number Prove that (an} converges to VA: (See Exercises 42-44 from Section Note that if we writc the an+1 above as (an)? 2an an+l = an we Obtain the formula for the Newton Raphson? method, uscd in calculus when approx- imating VA.'
Added by Kathryn V.
Step 1
We can do this by induction. For n = 1, we have a2 = a1/2 < a1 since a1 is positive. Now suppose that an < an+1 for some n. Then an+2 = (an+1)2 > an+1 > an, so the inequality holds for n+1 as well. Therefore, (an) is an increasing sequence. Show more…
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