Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergentto 0, then the series > (-1)"+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is then=1result still true?Consider the series defined by1111111 1(-1)"+1bn,-2438.4162nn=1where the sequence b, is defined by1 1 1 1 1 1 14' 3'8'4'16’5'32'1 1}n2n(a) Does this sequence {bn} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, andwhich does it fail?1(b) Show that this series diverges. (Hint: This series can also be written asSuppose that it didn2nn=1shouldconverge. If you add a certain geometric series to it, you're adding two convergent series together, so youget another convergent series but do you?)(c) Is the Alternating Series Test wrong? Explain why not.

(a) No, the sequence does not satisfy the assumptions of Alternating Series Test. It satisfies the…

Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent \n+1 to 0, then the series >(-1)n+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the n=1 result still true? Consider the series defined by 8. 1 1 + 16 1 1 1 - + 4 1 -- + 3 1 1 E(-1)"+1bn; - - .. 8. 4 n 2n n=1 where the sequence b, is defined by 1 1 1 1 1 1 1 1 1 1 2'4'3'8'4' 16'5' 32 2n n (a) Does this sequence {b„} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and which does it fail? 1 Σ E(;-). Suppose that it did (b) Show that this series diverges. (Hint: This series can also be written as 2n n=1 converge. If you add a certain geometric series to it, you're adding two convergent series together, so you should get another convergent series-but do you?) (c) Is the Alternating Series Test wrong? Explain why not.

Problem 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent \n+1 to 0, then the series >(-1)n+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the n=1 result still true? Consider the series defined by 8. 1 1 + 16 1 1 1 - + 4 1 -- + 3 1 1 E(-1)"+1bn; - - .. 8. 4 n 2n n=1 where the sequence b, is defined by 1 1 1 1 1 1 1 1 1 1 2'4'3'8'4' 16'5' 32 2n n (a) Does this sequence {b„} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and which does it fail? 1 Σ E(;-). Suppose that it did (b) Show that this series diverges. (Hint: This series can also be written as 2n n=1 converge. If you add a certain geometric series to it, you're adding two convergent series together, so you should get another convergent series-but do you?) (c) Is the Alternating Series Test wrong? Explain why not.

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 1. The Alternating Series Test states that if the positive sequence {b„} is (1) decreasing, and (2) convergent
to 0, then the series > (-1)"+'bn converges. But what if we drop the assumption that {bn} is decreasing? Is the
n=1
result still true?
Consider the series defined by
1
1
1
1
1
1
1 1
(-1)"+1bn,
-
2
4
3
8.
4
16
2n
n=1
where the sequence b, is defined by
1 1 1 1 1 1 1
4' 3'8'4'16’5'32'
1 1
}
n
2n
(a) Does this sequence {bn} satisfy the assumptions of the Alternating Series Test? Which does it satisfy, and
which does it fail?
1
(b) Show that this series diverges. (Hint: This series can also be written as
Suppose that it did
n
2n
n=1
should
converge. If you add a certain geometric series to it, you're adding two convergent series together, so you
get another convergent series but do you?)
(c) Is the Alternating Series Test wrong? Explain why not.

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