Pre-Calc Homework Solutions 370

Pre-Calc Homework Solutions 370 - 370 Section 9.1 49(a When...

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49. (a) When t 5 1, S 5 n 5 0 1 } 1 2 } 2 n 5 5 2. (b) S converges when ) } 1 1 t t } ) , 1, or ) t ) , ) 1 1 t ) . For t , 2 1, this inequality is equivalent to 2 t , 2 (1 1 t ), which is always false. For 2 1 # t , 0, the inequality is equivalent to 2 t , 1 1 t , which is true when t . 2 } 1 2 } . For t $ 0, the inequality is equivalent to t , 1 1 t , which is always true. Thus, S converges for all t . 2 } 1 2 } . (c) For t . 2 } 1 2 } , we have S 5 n 5 0 1 } 1 1 t t } 2 n 5 5 } (1 1 1 1 t ) t 2 t } 5 1 1 t , so S . 10 when t . 9. 50. (a) Comparing f ( t ) 5 } 1 1 4 t 2 } with } 1 2 a r } , the first term is a 5 4 and the common ratio is r 5 2 t 2 . First four terms: 4 2 4 t 2 1 4 t 4 2 4 t 6 General term: ( 2 1) n (4 t 2 n ) (b) Note that G (0) 5 0, so the constant term of the power series for G ( x ) will be 0. Integrate the terms for f ( x ) to obtain the terms for G ( x ). First four terms: 4 x 2 } 4 3 } x 3 1 } 4 5 } x 5 2 } 4 7 } x 7 General term: ( 2 1) n 1 } 2 n 4 1 1 } 2 x 2 n 1 1 (c) The series in part (a) converges when ) 2 t 2 ) , 1, so the interval of convergence is ( 2 1, 1).
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