(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

1. Prove that the sequence sqrt(n+1) - sqrt(n) converges to 0.

2. If sequence {an} is composed of real numbers and if lim as n goes to infinity of {a2n} = A and the limit as n goes to infinity of {a(2n-1)} = A, prove that {an} converges to 1. Is converse true?

3. Consider sequences {an} and {bn}, where bn = (an)^(1/n)

a. If {bn} converges to 1, does the sequence {an} necessarily converge?

b. If {bn} converges to 1, does the sequence {an} necessarily diverge?

c. does {bn} have to converge 1?

2. Relevant equations

3. The attempt at a solution

I'm not sure if I can divide sqrt(n) by sqrt(n) and prove that this new sequence goes to 1 without a loss of generality. As for the others, I am new to these proofs and any help would be much appreciated.

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# Homework Help: Some proofs of convergence

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