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Unformatted text preview: he recognizes that this limit is the number e.
The number e will be seen in Chapter 10.
6. This exercise concerns the sequence x n defined by the fact that x 1 1 and that, for each n
x n 1
a. 5 4x n 1 we have 2. Use Scientific Notebook to work out the first twenty terms in the sequence x n . b. Prove that 1 x n 2 for every n.
We use mathematical induction. Since x 1 1, the assertion p 1 is true. Now suppose that n is
any positive integer for which the assertion p n is true. We see that
1 5 2 41 5 4x n 2 5 42 2 2 and so the assertion p n1 is true. We deduce from mathematical induction that the assertion p n
is true for every positive integer n.
c. Prove that the sequence x n is strictly increasing.
We use mathematical induction. For each positive integer n we take p n to be the assertion that
x n x n1 . Since
x1 1 5 2 x2
we conclude that the assertion p 1 is true. Now suppose that n is any positive integer for which
the assertion p n happens to be true. Then
x n 2 5 4x n1 2 5 4x n 2 x n 1 and so the assertion p n1 must be true. We deduce from mathematical induction that the
assertion p n is true for every positive integer n.
d. Prove that the sequence has a limit x that satisfies the equation x 5
Notebook to make a 2D plot of the expression x5 172 4x 2 on the interval 4x 2 0. Ask Scientific
2, 2 and to solve the equation
4x 2 0 x5 x
1, 2
numerically. Compare the answer obtained here with the results that you obtained in part a.
7. a. Given that
for every number x 0, prove that f x
x 3. fx x 9
2x
2
3 for each n and that the equation f x 3 holds if and only if Solution: The desired result follows at once from the fact that whenever x 0 we have
fx x 9
2
2x
b. Given that x 1 4 and, for each n prove that the sequence x n x 3
2x 2 3. 1, we have x n 1 x n 9 ,
2x n
2
is decreasing and that the sequence converges to the number 3. Solution: Since x n1 f x n for each n and since f x 3 for every number x 3 we see at once
that x n 3 for every n. To see that x n is decreasing we observe that if n is any positive integer then
x2 9
xn 9
n
0.
x n x n 1 x n
2x n
2
2
Since the sequence x n is a decreasing sequence in the interval 3, Ý we know that x n is
convergent. If we write the limit of this sequence as x then it follows from the relationship
x n 1 x n 9
2x n
2
that
x x 9
2x
2
from which we deduce that x 3.
8. This exercise is a study of the sequence x n for which x 1 0 and
1
x n 1
2 xn
for every positive integer n. We note that this sequence is bounded below by 0 and above by 1/2.
a. Supply the definition
1
2x
to Scientific Notebook. Then open your Compute menu, click on Calculus, and choose to iterate the
function f ten times, starting at the number 0. Evaluate the column of numbers that you have obtained
accurately to ten decimal places and, in this way, show the first ten members of the sequence x n .
fx b. Show that
x n 2 2 x n
5 2x n
for every in...
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This note was uploaded on 11/26/2012 for the course MATH 2313 taught by Professor Staff during the Fall '08 term at Texas El Paso.
 Fall '08
 STAFF
 Math, Calculus

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