6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an index N such that an> 0 for all indices n > N.
6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an index N such that an> 0 for all indices n > N.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Solve number 6
![c. If the sequences {an + b} and {a} converge, then the sequence {bn} also
converges.
a.
d. If the sequence {lan]} converges, then the sequence {a} also converges.
2. Using only the Archimedean Property of R, give a direct e-N verification of the
following limits:
1
lim
n→∞ √√n
a.
/n
= 0
+
3. Using only the Archimedean Property of R, give a direct e-N verification of the
convergence of the following sequences:
n
+3
b.
n =
and show that an = 1/2.
d. Show that {an} does not converge.
j(j + 1)
2
b.
CONVERGENT SEQUENCES
4. For the sequence {an} defined in Example 2.3:
a. What are the terms a10, a20, 930?
b. Find the second index n for which an = 1/4 and the fourth index n for which
an = 1.
c. For j an odd natural number, set
+
Hint: Define a₁ =
1
= 0
lim
n→∞n +5
33
n²
n² +
j+1
2
5. For the sequence {an} defined in Example 2.3 and any rational number x in the
interval (0, 1], show that there are infinitely many indices n such that an = x.
6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an
index N such that an> 0 for all indices n > N.
7. Suppose that the sequence {an} converges to & and that the sequence {b} has the
property that there is an index N such that
lim n/n = 1.
84x
an = bn
for all indices n ≥ N.
Show that {b} also converges to l. (Suggestion: Use the Comparison Lemma for a
quick proof.)
8. Prove that the sequence {cn} converges to c if and only if the sequence {cn- c}
converges to 0.
9. Prove that the Archimedean Property of R is equivalent to the fact that
limn→∞ 1/n = 0.
10. Prove that
nl/n - 1 and use the Binomial Formula to show that for each](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F89f06fcb-a461-4580-b866-76a69e450ea3%2F24b6355b-4d8b-4ca7-817b-07b8280935fd%2Fjd792ur_processed.jpeg&w=3840&q=75)
Transcribed Image Text:c. If the sequences {an + b} and {a} converge, then the sequence {bn} also
converges.
a.
d. If the sequence {lan]} converges, then the sequence {a} also converges.
2. Using only the Archimedean Property of R, give a direct e-N verification of the
following limits:
1
lim
n→∞ √√n
a.
/n
= 0
+
3. Using only the Archimedean Property of R, give a direct e-N verification of the
convergence of the following sequences:
n
+3
b.
n =
and show that an = 1/2.
d. Show that {an} does not converge.
j(j + 1)
2
b.
CONVERGENT SEQUENCES
4. For the sequence {an} defined in Example 2.3:
a. What are the terms a10, a20, 930?
b. Find the second index n for which an = 1/4 and the fourth index n for which
an = 1.
c. For j an odd natural number, set
+
Hint: Define a₁ =
1
= 0
lim
n→∞n +5
33
n²
n² +
j+1
2
5. For the sequence {an} defined in Example 2.3 and any rational number x in the
interval (0, 1], show that there are infinitely many indices n such that an = x.
6. Suppose that the sequence {an} converges to a and that a > 0. Show that there is an
index N such that an> 0 for all indices n > N.
7. Suppose that the sequence {an} converges to & and that the sequence {b} has the
property that there is an index N such that
lim n/n = 1.
84x
an = bn
for all indices n ≥ N.
Show that {b} also converges to l. (Suggestion: Use the Comparison Lemma for a
quick proof.)
8. Prove that the sequence {cn} converges to c if and only if the sequence {cn- c}
converges to 0.
9. Prove that the Archimedean Property of R is equivalent to the fact that
limn→∞ 1/n = 0.
10. Prove that
nl/n - 1 and use the Binomial Formula to show that for each
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