Find bounding numbers for two interrelated sequences

[Entertainment Unit]

Homework Statement

Let $a$ and $b$ be positive numbers with $a \gt b$. Let $a_1$ be their arithmetic mean and $b_1$ their geometric mean:

$a_1 = \frac {a + b} 2$ and $b_1 = \sqrt{ab}$

Repeat this process so that, in general

$a_{n + 1} = \frac {a_n + b_n} 2$ and $b_{n + 1} = \sqrt{a_n b_n}$

(a) Use mathematical induction to show that $a_n \gt a_{n + 1} \gt b_{n + 1} \gt b_n$
(b) Deduce that both $\{a_n\}$ and $\{b_n\}$ are convergent.
(c) Show that $\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n$

Homework Equations

None that I'm aware of.

The Attempt at a Solution

I was able to complete (a) with some help.

I'm working on (b). I need to show that $\{a_n\}$ and $\{b_n\}$ are both bounded and monotonic in order to deduce that they are convergent. The sequences are both monotonic by the result of (a). I'm having trouble coming up with a bounding value for each of the following:

• $\{a_n\}$ (the decreasing sequence) is bounded below by ?
  
• $\{b_n\}$ (the increasing sequence) is bounded above by ?

to show that they are bounded. I think what's giving me the problem is the fact that the sequences are interrelated.

 

[PeroK]

What happens if $\{a_n\}$ is not bounded below?

 

[Entertainment Unit]

What happens if $\{a_n\}$ is not bounded below?

I think you're getting at $\lim_{n\to\infty} a_n$ would equal $-\infty$ and to calculate this limit and see what actually happens?

 

[PeroK]

I think you're getting at $\lim_{n\to\infty} a_n$ would equal $-\infty$ and to calculate this limit and see what actually happens?

Not really. What about this?

(a) Use mathematical induction to show that $a_n \gt a_{n + 1} \gt b_{n + 1} \gt b_n$

I was able to complete (a) with some help.

 

[Ray Vickson]

Homework Statement

Let $a$ and $b$ be positive numbers with $a \gt b$. Let $a_1$ be their arithmetic mean and $b_1$ their geometric mean:

$a_1 = \frac {a + b} 2$ and $b_1 = \sqrt{ab}$

Repeat this process so that, in general

$a_{n + 1} = \frac {a_n + b_n} 2$ and $b_{n + 1} = \sqrt{a_n b_n}$

(a) Use mathematical induction to show that $a_n \gt a_{n + 1} \gt b_{n + 1} \gt b_n$
(b) Deduce that both $\{a_n\}$ and $\{b_n\}$ are convergent.
(c) Show that $\lim_{n\to\infty} a_n = \lim_{n\to\infty} b_n$

Homework Equations

None that I'm aware of.

The Attempt at a Solution

I was able to complete (a) with some help.

I'm working on (b). I need to show that $\{a_n\}$ and $\{b_n\}$ are both bounded and monotonic in order to deduce that they are convergent. The sequences are both monotonic by the result of (a). I'm having trouble coming up with a bounding value for each of the following:

• $\{a_n\}$ (the decreasing sequence) is bounded below by ?
  
• $\{b_n\}$ (the increasing sequence) is bounded above by ?

to show that they are bounded. I think what's giving me the problem is the fact that the sequences are interrelated.

If you have proven that $a_n > a_{n + 1} > b_{n + 1} > b_n$ it follows that $b_1 < b_2 < b_3 < \cdots < a_3 < a_2 < a_1$.

 

[Entertainment Unit]

Not really. What about this?

Ok, I think I see what you were getting at now.

If $\{a_n\}$ were not bounded from below, it would run into $\{b_n\}$ at some point since $a_n > a_{n+1} > b_{n+1} > b_n$.

So, $\{a_n\}$ on the way down would run into $\{b_n\}$ on the way up as ${n\to\infty}$.

So the question becomes, at what value of $n$ do they meet (or at least get close since $a_n$ is strictly greater than $b_n$) so the bounds for both sequences can be calculated?

 

[Entertainment Unit]

If you have proven that $a_n > a_{n + 1} > b_{n + 1} > b_n$ it follows that $b_1 < b_2 < b_3 < \cdots < a_3 < a_2 < a_1$.

Yes, and by extension $b_1 < b_2 < b_3 < \cdots < b_n < b_{n + 1} < a_{n + 1} < a_n < \cdots < a_3 < a_2 < a_1$ which implies $b_1 < a_n$ and $b_n < a_1$.

It follows that, $\{a_n\}$ is bounded below by $b_1 = \sqrt{ab}$ and $\{b_n\}$ is bounded above by $a_1 = \frac {a + b} 2$.