Write a rigorous proof for the following statement. Statement: Let TeR be arbitrary. Then there exists a rearrangement of the alternating harmonic series (-1)*+1 Zn=1 s.t. the rearrangement converges to T. follow the road map I have provided below, make sure the proof is logically sound and that the logic is clear to the reader (not just you).
Write a rigorous proof for the following statement. Statement: Let TeR be arbitrary. Then there exists a rearrangement of the alternating harmonic series (-1)*+1 Zn=1 s.t. the rearrangement converges to T. follow the road map I have provided below, make sure the proof is logically sound and that the logic is clear to the reader (not just you).
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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Transcribed Image Text:Write a rigorous proof for the following statement.
Statement: Let TER be arbitrary. Then there exists a rearrangement of the alternating harmonic series
(-1)*+1
n
s.t. the rearrangement converges to T.
follow the road map I have provided below, make sure the
proof is logically sound and that the logic is clear to the reader (not just you).
1. Explain how you can rearrange the alternating harmonic series into the difference of two infinite
sequences =1 Cn and 1d, where cn, dn2 0. I have chosen c, to stand for credits and d, to stand
for debits. i.e.
2n=1
(-1)"+1
(C1 + C2 + C3 + ...) – (d1 + d2 + d3 + ...).
n
2. Suppose that the series E1 a, is conditionally convergent. Explain why the series -1 Cn and E-1 dn
both must diverge.
3. Pick any target value you would like to rearrange the series to converge to and denote it T.
4. Beginning with c, begin adding successive credits. Explain why the sum must eventually surpass T.
5. If you stop adding successive credits as soon as the sum surpasses T, how far from T can the sum be?
6. Now if you start subtracting successive debits starting at d1, explain why the sum must eventually
fall below T.
7. If you stop subtracting successive debits as soon as the sum falls below T, how far from T can the
sum be?
8. Explain how you can continue this process to rearrange the series so it has the form
(C1 + C2 + ... + Cn) – (d1 + d2 + d3 +
(Cn, +1 + Cn+2 + Cn; +3 + ... + Cn2) – (dm,+1 + dm,+2 + dm,+3 + ... + dm2) +
+ dm.) +
...
...
9. Explain why the rearrangement just created must have the sum T.
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