- #1
Entertainment Unit
- 16
- 1
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 ?