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Anton8ed10-8-63-soln - 51 SW[0&8 $9.7 Ami-57L vi.151...

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Unformatted text preview: +51 SW [0&8 $9.7) Ami-57L vi .151 great: 63. Prove: If the interval of convergence of the series 21°20 ck(x — 2:9)" is (3:0 — R. x0 + R], then the series Enn— verges conditionally at x0 + R. 00 O0 00 63. The assumption is that Z ckRk is convergent and Z Ck(—R)k is divergent. Suppose that Z ckR" ' Ic=o k=fl k=0 00 is absoluteiy convergent then Zea—Ry“ is also absolutely convergent and hence convergent k=0 00 because IckRki = [ck(—R)"E, which contradicts the assumption that Zea—R)" is divergent so Ic=e OD chRk must be conditionally convergent. Ic=0 ...
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