3100exam3

3100exam3 - MATH 3100 THIRD MIDTERM EXAM WITH SOLUTIONS...

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Unformatted text preview: MATH 3100 THIRD MIDTERM EXAM: WITH SOLUTIONS Directions: Please solve all the problems. Calculators are not permitted. You have 55 minutes. Good luck! 1) Let f ( x ) = x log x . a) Find the Taylor series expansion T ( x ) for f ( x ) centered at c = 1. Solution: Integrating the Taylor series equation 1 1- x = ∞ ∑ n =0 x n we get- log(1- x ) = ∞ ∑ n =0 x n +1 n + 1 , so log( x ) = log(1- (1- x )) = ∞ ∑ n =0- (1- x ) n +1 n + 1 = ∞ ∑ n =0 (- 1) n ( x- 1) n +1 n + 1 = ∞ ∑ n =1 (- 1) n- 1 ( x- 1) n n . We now want to multiply x but must be a little careful: multiplying a Taylor series in x by a power of x gives another Taylor series in x , but our Taylor series is centered at x- 1. So we write out the Taylor expansion of the polynomial x centered at c = 1, i.e., x = 1 + ( x- 1) and thus (1) f ( x ) = (1 + ( x- 1)) ∞ ∑ n =1 (- 1) n- 1 ( x- 1) n n = ∞ ∑ n =1 (- 1) n- 1 ( x- 1) n n + ∞ ∑ n =1 (- 1) n- 1 ( x- 1) n +1 n = ∞ ∑ n =1 (- 1) n- 1 ( x- 1) n n + ∞ ∑ n =2 (- 1) n ( x- 1) n n- 1 = ( x- 1) + ∞ ∑ n =2 ( (- 1) n- 1 n + (- 1) n n- 1 ) ( x- 1) n = ( x- 1) + ∞ ∑ n =2 (- 1) n ( x- 1) n n ( n- 1) . 1 b) For which values of x does the Taylor series T ( x ) converge?...
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3100exam3 - MATH 3100 THIRD MIDTERM EXAM WITH SOLUTIONS...

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