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GATE 2024 CH – Question 11

Engineering Mathematics · Calculus: Limits, continuity, differentiability, Taylor series and mean value theorem · 1 mark · Multiple choice

The first non-zero term in the Taylor series expansion of $(1 - x) - e^{-x}$ about $x = 0$ is

  1. 1
  2. −1
  3. $\frac{x^2}{2}$
  4. $-\frac{x^2}{2}$

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Correct answer: (D) $-\frac{x^2}{2}$

Explanation

Expand $e^{-x} = 1 - x + \frac{x^2}{2} - \cdots$. Then $(1 - x) - e^{-x} = -\frac{x^2}{2} + \frac{x^3}{6} - \cdots$. The constant and the linear terms cancel, so the first non-zero term is $-\frac{x^2}{2}$.