\"Pile of roots.\" Consider the sequence (an) = ( , , , ) (so and the subsequent
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Question
"Pile of roots." Consider the sequence (an) = ( , , , ) (so and the subsequent elements are defined by the recursive formula ) Prove that this sequence converges. Hint: start by proving that it is bounded. To do that simplify the first few terms of the sequence bringing them into the form an = 2[some power dependent on n] You should see a familiar pattern in the exponents. Use that information to provide a bound on an and proceed from there. Find the limit of this sequence. Hint: Call this limit L. Take the limit n rightarrow infinity on both sides of the equation and then solve the resulting equation for L.Explanation / Answer
a)
We should prove that the sequence is bounded.
a1 = 2 < 2
Assume that an-1 < 2, we have:
an = (2an-1) < (2*2) = 2
So for all values of n, an is bounded above by 2.
b)
lim an = L
So we have:
L = (2L) -> L^2 = 2L
L^2 - 2L = 0
L(L-2) = 0 -> L = 0 or 2 , but an is increasing, so L = 2.
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