hw97 - 9.7 Pages 487-488 1-19 odd 25 1 1 From the geometric...

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9.7 Pages 487-488 # 1-19 odd, 25 1. From the geometric series for 1 1 x with x replaced by –x, we get 2 3 4 5 1 1 1 x x x x x x = + + + + " , radius of convergence 1. 3. 2 2 3 1 1 1 2 ; , 1 (1 ) (1 ) (1 ) d d dx x dx x x x = = 3 1 so (1 ) x is 1 2 of the second derivative of 1 1 x . Thus, 2 3 3 1 1 3 6 10 (1 ) x x x x = + + + + " ; radius of convergence 1. 11. To get the series for 1 ln 1 x x + , we need to notice 1 ln ln(1 ) – ln(1– ) 1– x x x x + = + . 2 3 4 ln(1 ) , –1 1 2 3 4 x x x x x x + = + +… < 2 3 4 ln(1– ) , –1 1 2 3 4 x x x x x x = +… < 3 5 1 2 2 ln ln(1 ) ln(1 ) 2 1 3 5 x x x x x x x + = + = + + +⋅⋅⋅ radius of convergence 1. 13. Substitute –x for x in the series for x e to get: 2 3 4 5 1 2! 3! 4! 5! x x x x x e x = + + + " . 5. From the geometric series for 1 1 x with x replaced by 3 2 x , we get 2 3 2 3 1 1 1 3 9 27 2 1 3 2 3 2 2 4 8 1 2 1 3 9 27 2 4 8 16 x x x x x x x x = = + + + + = + + + + " " ;
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