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Explorations and Challenges 66. Rearranging series It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value 1,1 1 1 3 + I + ... = In 2. Show that by rearranging the terms (so the sign pattern is + + -), 1, =+= 1,1,1 2 5 7 4 ---In 2.

Explorations and Challenges 66. Rearranging series It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value 1,1 1 1 3 + I + ... = In 2. Show that by rearranging the terms (so the sign pattern is + + -), 1, =+= 1,1,1 2 5 7 4 ---In 2.

Advanced Engineering Mathematics
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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15 of 16 Explorations and Challenges 66. Rearranging series It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value 1 1 1-- + 1 + = ln 2. Show that by rearranging the terms (so the sign pattern is + + −), 11 1 1 1+=- +=+ --+ ... = -ln 2. 2 In 2.
Transcribed Image Text:15 of 16 Explorations and Challenges 66. Rearranging series It can be proved that if a series converges absolutely, then its terms may be summed in any order without changing the value of the series. However, if a series converges conditionally, then the value of the series depends on the order of summation. For example, the (conditionally convergent) alternating harmonic series has the value 1 1 1-- + 1 + = ln 2. Show that by rearranging the terms (so the sign pattern is + + −), 11 1 1 1+=- +=+ --+ ... = -ln 2. 2 In 2.
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