Problem 5. A sequence {bn} is a rearrangement of the sequence {an} if the terms of the sequences are the same, but appear in a different order. 8. (a) Suppose an absolutely converges, and {bn} is a rearrangement of the sequence {an}. Show that the sequence n=1 N of partial sums given by TN > bk| is bounded above. (Hint: Since > an is absolutely convergent, the k=1 n=1 sequence of partial sums SN = > Jak converges, and hence is bounded above by a number C > 0. For a fixed k=0 N, consider the finite sum TN made of a rearrangement of {|ak|}. Then the finite sum TN contains a term Jam], where M is the highest level term among the |ar|-terms in the finte sum TN. By filling in the remaining Jak| terms, show why TN < SM.) (b) Show that if > an absolutely converges, and {bn} is a rearrangement of the sequence {an}, then the series n=1 bn also converges absolutely. (Hint: Apply the Monotone Convergence Theorem to the sequence of partial n=1 sums, TN k=1 (c) Think about this result, and make an educated guess (not a mathematical argument) as to the following question: in the scenario of (a), are the series ). an and > bn in fact the same number?

Problem 5. A sequence {bn} is a rearrangement of the sequence {an} if the terms of the sequences are the same, but appear in a different order. 8. (a) Suppose an absolutely converges, and {bn} is a rearrangement of the sequence {an}. Show that the sequence n=1 N of partial sums given by TN > bk| is bounded above. (Hint: Since > an is absolutely convergent, the k=1 n=1 sequence of partial sums SN = > Jak converges, and hence is bounded above by a number C > 0. For a fixed k=0 N, consider the finite sum TN made of a rearrangement of {|ak|}. Then the finite sum TN contains a term Jam], where M is the highest level term among the |ar|-terms in the finte sum TN. By filling in the remaining Jak| terms, show why TN < SM.) (b) Show that if > an absolutely converges, and {bn} is a rearrangement of the sequence {an}, then the series n=1 bn also converges absolutely. (Hint: Apply the Monotone Convergence Theorem to the sequence of partial n=1 sums, TN k=1 (c) Think about this result, and make an educated guess (not a mathematical argument) as to the following question: in the scenario of (a), are the series ). an and > bn in fact the same number?

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Problem 5. A sequence {b,} is a rearrangement of the sequence {an} if the terms of the sequences are the same,
but appear in a different order.
(a) Suppose
an absolutely converges, and {bn} is a rearrangement of the sequence {an}. Show that the sequence
n=1
N
of partial sums given by TN
|bk| is bounded above. (Hint: Since ))
is absolutely convergent, the
an
k=1
n=1
N
sequence of partial sums SN
|ak| converges, and hence is bounded above by a number C > 0. For a fixed
k=0
N, consider the finite sum TN made of a rearrangement of {|ak }. Then the finite sum TN contains a term
Jam], where M is the highest level term among the |ak|-terms in the finte sum TN. By filling in the remaining
Jak| terms, show why TN < SM ·)
(b) Show that if > an absolutely converges, and {bn} is a rearrangement of the sequence {an}, then the series
n=1
b, also converges absolutely. (Hint: Apply the Monotone Convergence Theorem to the sequence of partial
n=1
sums, TN
bn.)
k=1
(c) Think about this result, and make an educated guess (not a mathematical argument) as to the following
question: in the scenario of (a), are the series ) an and ) bn in fact the same number?
n=1
n=1

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