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The Maclaurin series for \( \ln(1+x) \) is valid for

  • A.\( -1 < x \le 1 \)
  • B.all real \( x \) without restriction
  • C.\( |x| < 1/2 \) only
  • D.\( x \ge 0 \) only

Answer: A

The series \( x - x^{2}/2 + x^{3}/3 - \cdots \) has radius of convergence one. At \( x = 1 \) it becomes the alternating harmonic series, which converges to \( \ln 2 \); at \( x = -1 \) it is the harmonic series with a minus sign and diverges, matching \( \ln 0 \). So the interval is closed at the right and open at the left

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