MA 124 Exam 3 - Spring '09 Answers
- 1) Radius of Convergence = 1
Interval of Convergence (-1, 1).
- 2) \sum_{n=0}^infty (-1)^n 1/4^(n+1) x^(2n+1) for |x| < 2 .
- 3) a) c_n = f^(n)(0)/n! .
This is because if f(x) = a power series \sum_{n=0}^\infty c_n x^n , then
that series is the Maclaurin series, which has the above formula for c_n .
- 3) b) 30,240.
- 4) a) \sum_{n=0}^\infty (-1)^n ( -3 above n) x^n .
- 4) b) \sum_{n=0}^\infty (n+2)(n+1)/2 x^{n+2} .
- 4) c) 2
- 5) a) T_3(x) = x - 2x^2 + 2x^3 .
- 5) b) One can take M = 42 . Then |R_3(x)| < .00018 .
- 6) a) z = 2(cos(pi/6) + i sin(\pi/6)) , w = 2(cos(\pi/3) + i sin(\pi/3))
zw = 4i , z/w = (1/2) + i((\sqrt 3)/2)
- 6) b) 2^6(cos(\pi) + isin(\pi)) = -2^6 .
- 6) c) ((\sqrt 3)/2) + (1/2) i , (-(\sqrt 3)/2) + (1/2) i , -i .
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