Unformatted text preview: ( a ) ∞ X n = 1 ( x2 ) 2 n n , ( b ) ∞ X n = 1 (1 ) n √ n + 1 ( 2 x3 ) n . 4. (a) ( 7 points ) Find the ﬁrst four nonzero terms of the power series of f ( x ) = sin x around x = π 4 . (b) ( 7 points ) Find the power series of f ( x ) = x 2 1 + x around x = 0. 5. (a) ( 10 points ) Find the power series associated with Z x cos ( 2 t ) d t . (b) ( 10 points ) Obtain an approximation to Z 1 / 2 cos ( 2 x ) d x to less than 0.01 accuracy. (c) ( 4 points ) Find the exact value of the integral and verify that your approximation is valid up to the suggested accuracy. You may use sin ( 1 ) = 0.841471. 6. ( 14 points, 7 each ) Evaluate the limits using power series methods. ( a ) lim x → sin ( 2 x ) + 1e 2 x x 2 , ( b ) lim x → 1cosh x 1cos x . Good Luck!!!...
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 Winter '06
 LIM,JISUN
 Calculus, Power Series, Mathematical analysis, Radius of convergence, student ID number

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