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GATE 2024 BT – Question 29

Engineering Mathematics · Calculus: Sequences and series, tests for convergence · 1 mark · Multiple choice

The value of the series $1 + \sin x + \cos^2 x + \sin^3 x + \cdots$ at $x = \frac{\pi}{4}$ is ________.

  1. $\frac{1}{\sqrt2 + 1}$
  2. $\frac{\sqrt2}{\sqrt2 + 1}$
  3. $\frac{1}{\sqrt2 - 1}$
  4. $\frac{\sqrt2}{\sqrt2 - 1}$

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Correct answer: (D) $\frac{\sqrt2}{\sqrt2 - 1}$

Explanation

At $\pi/4$, $\sin x = \cos x = \frac{1}{\sqrt2} = r$, so the series is $1 + r + r^2 + r^3 + \cdots = \frac{1}{1 - r} = \frac{\sqrt2}{\sqrt2 - 1}$.