Suppose x_n in R, x_1=1, 2x_{n+1}=x_n+3/x_n. Then show that limxn exists and find its value.

Parker Bird 2022-07-15 Answered
Is this analysis problem or discrete math problem?
Suppose x n R , x 1 = 1 , 2 x n + 1 = x n + 3 / x n . Then show that limxn exists and find its value.
So is this problem (real) analysis problem or a discrete math one?
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Answers (1)

gutsy8v
Answered 2022-07-16 Author has 14 answers
Step 1
First, we will show that { x n } is strictly decreasing and bounded (not counting the first term). To do this, we must show that x n + 1 = x n 2 + 3 2 x n < x n ,, which is equivalent to x n > 3 x n ,, i.e. x n > 3 . This last statement we can prove easily (for n 2). By AM-GM, x n + 1 = x n 2 + 3 2 x n 3 ,, with equality iff x n = 3 .. Since x n 3 for n = 2 ,, equality never occurs, i.e. x n > 3 for all n 2 ,, from which we have 3 < x n + 1 < x n ,, as noted above.
Step 2
By the monotone convergence theorem, { x n } has a limit, say x. Then, by the given, 2 x = x + 3 x ,, from which x = 3 ..
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