homework5mat25

# homework5mat25 - n diverges Prove that the radius of...

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Homework 5 due Thursday September 4th 1. (a) Use induction to prove the following formula for all n 1: 1 3 + 1 15 + · · · + 1 4 n 2 - 1 = n 2 n + 1 (b) Find the sum n =1 1 4 n 2 - 1 . 2. Determine which of the following series converge and which diverge. Justify your answers. (a) cos( nπ/ 2). (b) 1 n ( n - 1) . (c) 3 n e - n . (d) 1 ln n . 3. (14.8) Assume a n > 0 and b n > 0 for all n . Show that if a n and b n converge, then a n b n converges. Hint: Show that a n b n a n + b n for all n . 4. (15.6) (a) Give an example of a divergent series a n so that a 2 n converges. (b) Prove that if a n converges with each a n > 0, then a 2 n also converges. (c) Give an example of a convergent series a n for which a 2 n diverges. 5. Find the radius of convergence of the following power series: (a) n ( x - 2) n . (b) n 3 n ! x n . 6. Suppose that the sequence ( a n ) is bounded but the series
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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 re-ordering of the natural numbers. A series ∑ a n is unconditionally convergent if ∑ a σ ( n ) converges for all σ . (In other words all re-orderings 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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