Let {an} be the sequence defined recursively by a1 = 1 andan+1 = 12 [an + (3/an)] for n

Chapter 9, Problem 28

(choose chapter or problem)

Let \(\left\{a_{n}\right\}\) be the sequence defined recursively by \(a_{1}=1\) and \(a_{n+1}=\frac{1}{2}\left[a_{n}+\left(3 / a_{n}\right)\right]\) for \(n \geq 1\)

(a) Show that \(a_{n} \geq \sqrt{3}\) for \(n \geq 2\) . [Hint: What is the minimum value of \(\frac{1}{2}[x+(3 / x)]\) for \(x>0\) ?]

(b) Show that \(\left\{a_{n}\right\}\) is eventually decreasing. [Hint: Examinc \(a_{n+1}-a_{n}\) or \(a_{n+1} / a_{n}\) and use the result in part (a).]

(c) Show that \(\left\{a_{m}\right\}\) converges and find its limit \(L\).

Equation Transcription:

Text Transcription:

{a_n}

a_1 = 1

a_n+1 = 1/2 [a_n + (3/a_n)] for n geq 1

a_n geq sqrt 3 for n geq 2

1/2 [x+(3/x)] for x > 0

a_n+1 - a_n

a_n+1 /a_n

{a_m}

L