The GATE Grind

GATE 2024 MT – Question 12

Engineering Mathematics · Calculus and vector calculus · 1 mark · Multiple choice

Which Taylor series represents $\ln[(1+x)/(1-x)]$ about the origin for real $|x|<1$?

  1. $x-x^2/2+x^3/3-\cdots$
  2. $2(x-x^2/2+x^3/3-\cdots)$
  3. $x+x^3/3+x^5/5+\cdots$
  4. $2(x+x^3/3+x^5/5+\cdots)$

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Correct answer: (D) $2(x+x^3/3+x^5/5+\cdots)$

Explanation

Write the expression as ln(1+x)−ln(1−x).

Subtracting their Taylor expansions cancels even powers and doubles odd powers, giving **2(x+x³/3+x⁵/5+…), D**, valid for |x|<1.