Let {an} be a sequence defined recursively by a1 = 1 and an+1 = 2√ 1...

Let {an} be a sequence defined recursively by a1 = 1 and an+1 = 2√ 1 + an where n ∈ N

(b) Does {an} converge or diverge? Justify your answer, making sure to cite appropriate hypotheses/theorem(s) used. Hint : Try BMCT [WHY?].

(c) (Challenge) If {an} converges then find its limit. Make sure to fully justify your answer.

Expert Solution

milcah answered 2 years ago

Write the first four terms of the sequence defined by the recursive formula a1 = 2, an = an − 1 + n.

Let a1 ≥ a2, . . . , an be a sequence of positive integers whose...

Let a1 ≥ a2, . . . , an be a sequence of positive integers whose sum is 2n − 2. Prove that there exists a tree T on n vertices whose vertices have degrees a1, a2, . . . , an. Sketch of solution: Prove that there exist i and j such that ai = 1 and aj ≥ 2. Remove ai, subtract 1 from aj and induct on n.

Define a sequence from R as follows. Fix r > 1. Let a1 = 1 and...

Define a sequence from R as follows. Fix r > 1. Let a1 = 1 and define recursively, an+1 = (1/r) (an + r + 1). Show, by induction, that (an) is increasing and bounded above by (r+1)/(r−1) . Does the sequence converge?

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Question in graph theory: 1. Let (a1,a2,a3,...an) be a sequence of integers. Given that the sum...

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1.Prove the following statements: . (a) If bn is recursively defined by bn =bn−1+3 for all...

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