# q3s - ln(1-t = Z-1 1-t dt = Z ∞ X n =0-t n dt = ∞ X n...

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MATH 153 Quiz 3 Solutions Name: 1. [12 points] Find the Maclaurin series for f ( x ) using the deﬁnition of a Maclaurin series. Do not show that R n ( x ) 0. Also ﬁnd the associated radius of convergence. f ( x ) = sin(2 x ) First, we ﬁnd the pattern in the derivatives: f (0) ( x ) = f ( x ) = sin(2 x ) f (0) = 0 f 0 ( x ) = 2 cos(2 x ) f 0 (0) = 2 f 00 ( x ) = 2 2 ( - sin(2 x )) f 00 (0) = 0 f 000 ( x ) = 2 3 ( - cos(2 x )) f 000 (0) = - 8 Putting these into a power series gives f ( x ) = 0 0! x 0 + 2 1! x 1 + 0 2! x 2 = - 8 3! x 3 + ..., or without the even terms, f ( x ) = X n =0 ( - 1) n 2 2 n +1 (2 n + 1)! x 2 n +1 . To ﬁnd the radius of convergence, one applies the Ratio Test: lim n →∞ ± ± ± ± ± ± ( - 1) n +1 2 2 n +3 x 2 n +3 (2 n +3)! ( - 1) n 2 2 n +1 x 2 n +1 (2 n +1)! ± ± ± ± ± ± = lim n →∞ 2 2 n +3 | x | 2 n +3 (2 n + 3)! · (2 n + 1)! 2 2 n +1 | x | 2 n +1 = lim n →∞ 2 2 | x | 2 (2 n + 2)(2 n + 3) = 0 , since for any x this is a constant over a quadratic. Thus, since 0 < 1, this series converges for all x ; that is, R = .

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2. [8 points] Evaluate the indeﬁnite integral as a power series. What is the radius of convergence? Z t ln(1 - t ) dt We ﬁrst ﬁnd a power series for ln(1 - t ) by ﬁnding one for its derivative: d dx ln(1 - t ) = - 1 1 - t = X n =0 - t n . This forces
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Unformatted text preview: ln(1-t ) = Z-1 1-t dt = Z ∞ X n =0-t n dt = ∞ X n =0 Z-t n dt = ∞ X n =0-t n +1 n + 1 + C. Plugging in t = 0 gives 0 = ln(1-0) = ∞ X n =0 0 + C, so that C = 0. Putting it all together, we’ve so far discovered that ln(1-t ) = ∞ X n =0-t n +1 n + 1 . Finally, multiplying by t and integrating yields Z t ln(1-t ) dt = Z t ∞ X n =0-t n +1 n + 1 dt = Z ∞ X n =0-t n +2 n + 1 dt = ∞ X n =0 Z-t n +2 n + 1 dt = ∞ X n =0-t n +3 ( n + 1)( n + 3) + C. Finally, since the radius of convergence of our ﬁrst series ( ∑ ∞ n =0-t n ) was 1, and we only changed it in ways (integration, multiplication by t ) that don’t aﬀect the radius of convergence, our ﬁnal radius is also R = 1. Alternatively, you can check this using Ratio Test if you are so inclined. I’m not....
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q3s - ln(1-t = Z-1 1-t dt = Z ∞ X n =0-t n dt = ∞ X n...

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