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Unformatted text preview: n 1 xn 1 1 1 1
n1 j2 1 2 1
n j
n n 11
j2 1 2
1
n1 j1
n1 1 x n 1 .
j! d. Deduce that the sequence x n converges to a number between 2 and 3. Have you seen this number
before?
The student is being asked informally whether he/she recognizes that this limit is the number e.
The number e will be seen in Chapter 10. 147 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
4x 2 on the interval
Notebook to make a 2D plot of the expression
equation
x 5 4x 2 0
x5 4x 2 0. Ask Scientific
2, 2 and to solve the 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 148 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...
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 Fall '08
 STAFF
 Math, Calculus

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