MA 124 , Fall-07, Exam 3 Answers

---

• 1) Radius of convergence is infinity.
  Interval of convergence is (-infinity, infinity).
• 2) a) Substitute (-x) for x, then integrate to get:
  C + \sum-{n=0}^\infty (-1)^n x^{n+1}/(n+1) . Setting x = 0 gives C = 0 and hence ln(1+x) = \sum-{n=0}^\infty (-1)^n x^{n+1}/(n+1).
• 2) b) We can multiply the series in a) by 2x to get
  2x*ln(1+x) = \sum-{n=0}^\infty (-1)^n 2 x^{n+2}/(n+1).
• 2) c) We can substitute 4x^2 for x in a). Since (4x^2)^(n+1) = 4^{n+1}*x^{2n+2} we get:
  ln(1 + 4x^2) = \sum-{n=0}^\infty (-1)^n 4^{n+1} x^{2n+2}/(n+1).
• 2 d) The radius of convergence is 1 for a) and b). The radius of convergence is 1/2 for c). (Since substituting 4x^2 for x gives |4x^2| < 1 which implies |x| < 1/2 ).
• 3) By calculating f^(n)(\pi/2) for n = 0, 1, 2, 3 and observing that the pattern repeats after that we see only have even terms 2n, and those alternate with pattern (-1)^n, so the coefficent of (x-pi/2)^{2n} is (-1)^n/(2n)! hence the Taylor series is:
  \sum_{n=0}^\infty 1/(2n)! (x - pi/2)^{2n}
• 4) Use the binomial series formula with k = -4 . Computing the binomial coefficients (-4 \above n) we get pattern
  (-4 \above n) = (-1)^n (4 * 5 * .... * (n+3))/n! OR = (-1)^n ((n+1)(n+2)(n+3))/6 so
  1/(1+x)^4 = \sum_{n=0}^\infty (-1)^n ((n+1)(n+2)(n+3))/6 x^n
• 5 a) Computing Taylor coefficents we get:
  T_3(x) = 2 + (1/4)(x - 4) - (1/64)(x - 4)^2 + (1/512)(x - 4)^3 .
• 5) b) Computing f^(4)(x) one sees it has largest value in the interval [3.8, 4.2] when x = 3.8. and then |f^(4)(x)| \leq |f^(4)(3.8)| < .009 (rounding UP to one significant digit). Taking M = .009 and using |x - 4| \leq .2 for x in the interval [3.8, 4.2] one gets
  |R_3(x)| < (.009)(.2)^4/4! < 6 x 10^(-7) = .0000006.
• 6a) 5 + 12i , |z| = 13
• 6 b) -1/4 +/- (\sqrt(3)/4)i
• 6 c) 2^10 (cos \pi + i sin \pi) in polar form. -2^10 in a + bi form ,
• 6 d) log(i^4) = log(1) = {i(n(2\pi)) | n in Z}
  4log(i) = { i(2\pi + n(8\pi)) | n in Z}
  The are NOT the same i.e. i4\pi is in the first set, but not in the second set.

  ---

  IF YOU HAVE FURTHER QUESTIONS ABOUT HOW TO DO THESE PROBLEMS I WILL BE HAPPY TO DISCUSS THEM WITH YOU DURING MY OFFICE HOURS OR AT MATH HELP.

  ---

  Return to Diane Meuser's home page