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Let 0 < θ < 1 and let (xn) be a sequence where |xn+1 − xn|...

Let 0 < θ < 1 and let (xn) be a sequence where |xn+1 − xn| ≤ θn  for n = 1, 2, . . ..

a) Show that for any 1 ≤ n < m one has |xm − xn| ≤ (θn/ 1-θ )*(1 − θ m−n ).

Conclude that (xn) is Cauchy

b)If lim xn = x* , prove the following error in approximation (the "error in approximation" is the same as error estimation in Taylor Theorem) in t: |x ∗ − xn| ≤ θ n /1 − θ , n = 1, 2, . . . .

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