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# hw6 - 4 Use power series to approximate Z 1 1 1 x 5 dx...

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Calculus II, Homework 6 (Last homework!!!) November 27, 2007 1. Find the radius and interval of convergence of each of the following series (a) n 3 x n (b) 2 n n ! x n (c) 2 n n 2 ( x - 1) n (d) n 3 3 n ( x + 3) n 2. True or false. If true, briefly explain why, if false, provide a specific counterexample (a) There exists a power series with interval of convergence [0 , ). (b) If c n 4 n converges then c n ( - 2) n converges. (c) If c n 4 n converges then c n ( - 4) n converges. (d) If a n > 0 for all n and a n converges then ( - 1) n a n converges. (e) 0 . 9999999 ... = 1 3. Find a power series representation of 3 x
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Unformatted text preview: 4. Use power series to approximate Z . 1 1 1 + x 5 dx correct to within 10-5 of its true value. 5. Deﬁne J ( x ) = ∞ X n =0 (-1) n 2 2 n x 2 n ( n !) 2 (a) What is the radius of convergence of J ? (b) What is the interval of convergence of J ? (c) Show that J satisﬁes the diﬀerential equation x 2 J 00 ( x ) + xJ ( x ) + x 2 J ( x ) = 0 . 6. Find the taylor series of (a) cos( x 2 ) (b) xe-x (c) x ln(1-x 2 ) 1...
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