Write a rigorous proof for the following statement.Statement: Let TER be arbitrary. Then there exists a rearrangement of the alternating harmonic series(-1)*+1ns.t. the rearrangement converges to T.follow the road map I have provided below, make sure theproof 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 infinitesequences =1 Cn and 1d, where cn, dn2 0. I have chosen c, to stand for credits and d, to standfor debits. i.e.2n=1(-1)"+1(C1 + C2 + C3 + ...) – (d1 + d2 + d3 + ...).n2. Suppose that the series E1 a, is conditionally convergent. Explain why the series -1 Cn and E-1 dnboth 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 eventuallyfall below T.7. If you stop subtracting successive debits as soon as the sum falls below T, how far from T can thesum 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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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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