i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If SnX1 X2 + x3 = ... • (-1)n-¹ xn prove that the sequence {sn} converges using the Cauchy criterion. [Hint: You need to prove that {sn} if m > n Sm Sn = (-1)"xn+1 - ... + (−1) m−¹ xm = (−1)″ (xn+1 + ... + (−1)m-n−¹xn), then use the associative property of the sum and the fact that {n} is monotone decreasing and în → 0.]
i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If SnX1 X2 + x3 = ... • (-1)n-¹ xn prove that the sequence {sn} converges using the Cauchy criterion. [Hint: You need to prove that {sn} if m > n Sm Sn = (-1)"xn+1 - ... + (−1) m−¹ xm = (−1)″ (xn+1 + ... + (−1)m-n−¹xn), then use the associative property of the sum and the fact that {n} is monotone decreasing and în → 0.]
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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Prove the statement
![i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If
SnX1 X2 + x3 = ... • (-1)n-¹ xn
prove that the sequence {sn} converges using the Cauchy criterion.
[Hint: You need to prove that {sn} if m > n
Sm Sn = (-1)"xn+1 - ... + (−1) m−¹ xm = (−1)″ (xn+1 + ... + (−1)m-n−¹xn),
then use the associative property of the sum and the fact that {n} is monotone decreasing and în → 0.]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfba8c1b-379a-495a-9284-26414a9f3892%2F83077edd-a0c6-466d-b835-f30bdfbfe5cf%2F4g2n1q2_processed.png&w=3840&q=75)
Transcribed Image Text:i) Let {n} a sequence of non-negative terms that is monotone decreasing and în → 0. If
SnX1 X2 + x3 = ... • (-1)n-¹ xn
prove that the sequence {sn} converges using the Cauchy criterion.
[Hint: You need to prove that {sn} if m > n
Sm Sn = (-1)"xn+1 - ... + (−1) m−¹ xm = (−1)″ (xn+1 + ... + (−1)m-n−¹xn),
then use the associative property of the sum and the fact that {n} is monotone decreasing and în → 0.]
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