Unformatted text preview: n diverges. Prove that the radius of convergence of the power series ∑ a n x n is equal to 1. 7. Let σ : N → N be a bijection. σ is a permutation, or reordering of the natural numbers. A series ∑ a n is unconditionally convergent if ∑ a σ ( n ) converges for all σ . (In other words all reorderings of the series converge.) (a) Show by example that not all series are unconditionally convergent. Start with ∑ (1) n n , which we know converges, and call its limit s . Reorder the terms to make a series that converges to 1 2 s . (b) Prove Dirchlet’s theorem which says that absolutely convergent series are unconditionally convergent. 1...
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 Winter '11
 Wiley
 Differential Equations, Equations

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