prove or desprove that if An->1 then An+1/An ->1 ??
T transgalactic Junior Member Joined Nov 20, 2008 Messages 58 Dec 3, 2008 #1 prove or desprove that if An->1 then An+1/An ->1 ??
P PAULK Junior Member Joined Dec 13, 2007 Messages 124 Dec 3, 2008 #2 Try something like this: If An --> 1, then given e, exists N s.t. | An - 1 | < e for all n > N. Now take: |A(n+1)/A(n) - 1 | = A(n+1) - A(n) ------------- A(n) Assume that | A(n) - 1 | < 1/2 Then A(n) > 1/2 A(n+1) - A(n) | ------------- | < 2 | A(n + 1) - A(n) | A(n) Now choose N s.t. | A(n+1) - 1| < e/4 and | A(n) - 1| < e/4 Then |A(n+1) - A(n) | < e/2 and A(n+1) - A(n) | ------------- | < 2 | A(n + 1) - A(n) | < 2(e/2) = e A(n)
Try something like this: If An --> 1, then given e, exists N s.t. | An - 1 | < e for all n > N. Now take: |A(n+1)/A(n) - 1 | = A(n+1) - A(n) ------------- A(n) Assume that | A(n) - 1 | < 1/2 Then A(n) > 1/2 A(n+1) - A(n) | ------------- | < 2 | A(n + 1) - A(n) | A(n) Now choose N s.t. | A(n+1) - 1| < e/4 and | A(n) - 1| < e/4 Then |A(n+1) - A(n) | < e/2 and A(n+1) - A(n) | ------------- | < 2 | A(n + 1) - A(n) | < 2(e/2) = e A(n)
