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# 119 - radius of convergence Z ln(1-t t dt Solution Note...

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MATH 153 Selected Solutions for § 11.9 Exercise 8 : Find a power series representation for the function and determine the interval of convergence. f ( x ) = x 2 x 2 + 1 Solution : Note that we can rewrite the function as x 1+2 x 2 , so that it looks like the sum of a geometric series with ﬁrst term a = x and ratio r = - 2 x 2 . So we write it as such: x 2 x 2 + 1 = X n =0 x ( - 2 x 2 ) n . We want this to look more like c n x n , so we rewrite it a bit to get the ﬁnal answer: x 2 x 2 + 1 = X n =0 ( - 2) n x 2 n +1 . As the series is geometric, it converges exactly when | r | = | - 2 x 2 | < 1, or when | x | < 1 / 2. Thus, the interval of convergence is ( - 1 / 2 , 1 / 2). Exercise 16 : Find a power series representation for the function and deter- mine the radius convergence. f ( x ) = x 2 (1 - 2 x ) 2 Solution : Note that 1 (1 - 2 x ) 2 = d dx 1 2 1 1 - 2 x . This is half the sum of a geometric series with a = 1 and r = 2 x ; thus, 1 (1 - 2 x ) 2 = d dx 1 2 1 1 - 2 x = d dx 1 2 X n =0 (2 x ) n = 1 2 X n =0 d dx (2 x ) n = X n =1 1 2 n 2 n x n - 1 = X n =1 n 2 n - 1 x n - 1 . So, multiplying by x 2 gives us x 2 (1 - 2 x ) 2 = X n =1 n 2 n - 1 x n +1 . 1

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2 Exercise 24 : Evaluate the indeﬁnite integral as a power series. What is the
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Unformatted text preview: radius of convergence? Z ln(1-t ) t dt Solution : Note that d dt ln(1-t ) =-1 1-t , which is the sum of the geometric series ∑ ∞ n =0-t n . Thus, we have that ln(1-t ) = Z ∞ X n =0-t n dt = ∞ X n =0 Z-t n dt = ∞ X n =0-1 n + 1 t n +1 + C. Plugging in t = 0 yields ln(1) = 0 = ∑ 0 + C , so that C = 0 and ln(1-t ) = ∑ ∞ n =0-1 n +1 t n +1 . Finally, dividing by t yields ln(1-t ) t = ∞ X n =0-1 n + 1 t n . We can integrate this to get Z ln(1-t ) t dt = Z ∞ X n =0-1 n + 1 t n dt = ∞ X n =0 Z-1 n + 1 t n dt = ∞ X n =0-1 ( n + 1) 2 t n +1 + C. The radius of convergence of the very ﬁrst series we created is 1 (a geometric series with ratio t ), and none of our steps do anything to change that. Thus, our ﬁnal radius of convergence is also 1....
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119 - radius of convergence Z ln(1-t t dt Solution Note...

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