LETU MATH 2013 - Power Series (8 pages)

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Power Series



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Power Series Lesson 8 7 Definition A power series in x c has the form a x c k k a0 a1 x c a1 x c k 1 Consider this as an extension of a polynomial in x 2 Convergence of Power Series k ak x For the power series k 0 exactly one of the following is true 1 The series converges for all x 2 The series converges only for x 0 3 The series converges absolutely for all x in R R diverges for x R may converge or diverge at R or R Example xk k 1 k Consider the power series What happens at x 0 Use generalized ratio test for x 0 Try this k x k 1 k x k 1 k 1 L lim k k x k Dealing with Endpoints Consider k 1 x k k 1 Converges trivially at x 0 Use ratio test k 1 lim k k x k k 1 x Limit x converges when x 1 Interval of convergence 1 x 1 Dealing with Endpoints Now what about when x 1 k 1 k k 1 1 k 1 1 k k 1 At x 1 diverges by the divergence test At x 1 also diverges by divergence test Final conclusion convergence set is 1 1 Try Another Consider k k 1 2 x 2 k 3k Again use ratio test 1 Should get L x 2 3 which must be 1 or 1 x 5 Now check the endpoints 1 and 5 Power Assignment Lesson 8 7 Page 552 Exercises 1 25 EOO


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