exercises_13 - (a) X n =1 (-1) n 2-n ( n 2-1) n 2 + 1 . (b)...

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Analysis 1: Exercises 13 1. Let p > 1 and x > y > 0. Use the Mean Value Theorem to prove the inequality py p - 1 ( x - y ) x p - y p px p - 1 ( x - y ) . 2. Let m,n N . Use l’Hopital’s rule to compute the limits ( a ) lim x 1 x n - 1 x 2 - x , ( b ) lim x 1 2 (1 - x ) m - x m (1 - x ) n - x n , ( c ) lim x 1 nx n +2 - ( n + 1) x n +1 + x ( x - 1) 2 . 3. Write the following polynomials in x as polynomials in ( x - 3). (a) x 2 - 4 x - 9, (b) x 5 . 4. Write down the Taylor expansion at a up to the term in x 2 with remainder in Lagrange form for the following functions. (a) f ( x ) = 1 1 + x ; a = 0. (b) f ( x ) = 1 + x ; a = 0. 5. Prove that, if f 00 ( a ) exists, then f 00 ( a ) = lim h 0 f ( a + h ) + f ( a - h ) - 2 f ( a ) h 2 . 6. Determine whether or not the following series are absolutely convergent.
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Unformatted text preview: (a) X n =1 (-1) n 2-n ( n 2-1) n 2 + 1 . (b) X n =1 (-1) n n + 1- n n . (c) X n =1 x n n ! , where x is some real number. 7. Let X n =1 a n be a convergent series of positive terms. Prove that (a) if ( n N )( | b n | a n ), then X n =1 b n is absolutely convergent; 2 (b) X n =1 a n x n is absolutely convergent for-1 x 1....
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exercises_13 - (a) X n =1 (-1) n 2-n ( n 2-1) n 2 + 1 . (b)...

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