quizsol12

# quizsol12 - , for-1 &amp;amp;lt; x &amp;amp;lt; 1 ,...

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Math 104, Solution to Quiz 12 Instructor: Guoliang Wu August 10, 2009 1. (15 points) Prove that | sin x - sin y | ≤ | x - y | , for all x, y R . Proof. If x = y , then obviously | sin x - sin y | = | x - y | = 0 . If x 6 = y , by the mean value theorem , there exists x 0 between x and y such that sin x - sin y = sin 0 x 0 ( x - y ) ⇒ | sin x - sin y | = | cos x 0 || x - y | ≤ | x - y | . In either case, we have the desired inequality. 2. (a) (10 points) Derive an explicit formula for the power series X n =1 nx n +1 . Hint: You may use the formula 1 1 - x = X n =0 x n , for - 1 < x < 1 . Solution: Since 1 1 - x = n =0 x n

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Unformatted text preview: , for-1 &lt; x &lt; 1 , differentiating the series, we obtain that 1 (1-x ) 2 = X n =1 nx n-1 ,-1 &lt; x &lt; 1 . Then multiplying by x 2 , f ( x ) := x 2 (1-x ) 2 = X n =1 nx n +1 ,-1 &lt; x &lt; 1 . 1 (b) (5 points) Find the sum of the series X n =1 (-1) n n 2 n +1 . Solution: Note that X n =1 (-1) n n 2 n +1 =- X n =1 (-1) n +1 n 2 n +1 =-f -1 2 =-1 / 4 9 / 4 =-1 9 . 2...
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## quizsol12 - , for-1 &amp;amp;lt; x &amp;amp;lt; 1 ,...

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