frsol - Final Review Problems November 27, 2007 1. Given f...

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Unformatted text preview: Final Review Problems November 27, 2007 1. Given f ( x ) = X n =0 a n ( x- c ) n , let R = 1 lim n n p | a n | . Explain why f converges absolutely for all x such that | x- c | < R and diverges for all x such that | x- c | > R . Solution: Put L = 1 lim n p | a n || x- c | n = 1 (lim n p | a n | ) | x- c | . By the root test, if L < 1 (ie | x- c | < R ) then f ( x ) converges absolutely. Again by the root test, if L > 1, (ie | x- c | > R ) then f ( x ) diverges. 2. Let p, q be real numbers with p < q . Find a power series whose interval of convergence is (a) ( p, q ) (b) ( p, q ] (c) [ p, q ) (d) [ p, q ] Solution: a) ( 2 q- p ) n ( x- ( p + q 2 )) n b) ( 2 q- p ) n (- 1) n n ( x- ( p + q 2 )) n c) ( 2 q- p ) n 1 n ( x- ( p + q 2 )) n d) ( 2 q- p ) n 1 n 2 ( x- ( p + q 2 )) n 3. Suppose c n x n converges when x =- 4 and diverges when x = 6. What can be said about the convergence or divergence of (a) c n (b) c n 8 n (c) c n (- 3) n (d) (- 1) n c n 9 n Let I be the interval of convergence of f ( x ) = c n x n . Notice that [- 4 , 4) I [- 6 , 6). So a) f (1) = c n converges (absolutely). b) f (8) = c n 8 n diverges. c) f (- 3) = c n (- 3) n converges. d) f (- 9) = (- 1) n c n 9 n diverges. 4. Suppose c n x n has radius of convergence 2 and d n x n has radius of convergence 3. What is the radius of convergence of ( c n + d n ) x n ? Notice that f ( x ) = c n x n converges on (- 2 , 2) (though its interval of convergence may be larger) and g ( x ) = d n x n converges on (- 3 , 3) (though its interval of convergence may be larger). Therefore ( f + g )( x ) = ( c n + d n ) x n converges on (- 2 , 2). Letting the radius of convergence of f + g to be R , this shows that R 2. Claim: R 2. If not, then f + g would converge for all sufficiently small 0...
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frsol - Final Review Problems November 27, 2007 1. Given f...

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