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7. Let $a_1 = \sqrt{2}$ and let $a_n$ for $n \ge 2$ be defined recursively by the formula $a_{n+1} = \sqrt{2 + \sqrt{a_n}}$. \begin{itemize} \item Prove that $a_n$ is bounded above by 2. Hint: Use induction on $n$. (4 pts) \item Show that {$a_n$} is convergent and find its limit. Hint: Use induction to show the sequence is monotonic. Then use the Monotone Convergence Theorem to show convergence. To find the limit, use the technique you use in the previous problem. (6 pts) \end{itemize}

          7. Let $a_1 = \sqrt{2}$ and let $a_n$ for $n \ge 2$ be defined recursively by the formula
$a_{n+1} = \sqrt{2 + \sqrt{a_n}}$.
\begin{itemize}
    \item Prove that $a_n$ is bounded above by 2. Hint: Use induction on $n$. (4 pts)
    \item Show that {$a_n$} is convergent and find its limit. Hint: Use induction to show the sequence
is monotonic. Then use the Monotone Convergence Theorem to show convergence. To
find the limit, use the technique you use in the previous problem. (6 pts)
\end{itemize}

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7. Let a1 = √(2) and let an for n ≥ 2 be defined recursively by the formula
an+1 = √(2 + √(an)).

* Prove that an is bounded above by 2. Hint: Use induction on n. (4 pts)

* Show that an is convergent and find its limit. Hint: Use induction to show the sequence
is monotonic. Then use the Monotone Convergence Theorem to show convergence. To
find the limit, use the technique you use in the previous problem. (6 pts)

Added by Juan Manuel D.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Uma Kumari

08/18/2021

Step 1

Base case (n=1): a_1 = sqrt(2) < 2, so the base case holds. Inductive step: Assume that a_k ≤ 2 for some k ≥ 1. We want to show that a_k+1 ≤ 2. a_k+1 = sqrt(2 + a_k) Since a_k ≤ 2, we have 2 + a_k ≤ 4, and thus sqrt(2 + a_k) ≤ sqrt(4) = 2. So, a_k+1 ≤ 2, Show more…

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Let a_1=sq rt 2 and let a_n for n>=2 be defined recursively by the formula a_n+1=sq rt 2 + sq rt a_n. Prove that a_n is bounded above by 2. Hint: use induction on 2. Show that a_n is convergent and find it's limit. Hint: Use induction to show its monotonic. Then use the monotone convergence theorem to show convergence. To find the limit, use the technique you used in the previous problem. 7.Let a=2 and let an for n2 be defined recursively by the formula an+1=V2+an. Prove that an is bounded above by 2.Hint: Use induction on n.4 pts Show that {an} is convergent and find its limit.Hint: Use induction to show the sequence is monotonic. Then use the Monotone Convergence Theorem to show convergence. To find the limit,use the technique you use in the previous problem.6 pts)

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