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GATE 2016 EE – Question 37

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 2 marks · Numerical answer

Let $S = \sum_{n=0}^{\infty} n\alpha^n$ where $|\alpha| < 1$. The value of $\alpha$ in the range $0 < \alpha < 1$, such that $S = 2\alpha$ is ________.

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Correct answer: 0.28 to 0.31

Explanation

The series sums to $S = \frac{\alpha}{(1 - \alpha)^2}$. Setting $S = 2\alpha$ gives $\frac{1}{(1 - \alpha)^2} = 2$, so $1 - \alpha = \frac{1}{\sqrt{2}}$ and $\alpha = 1 - 0.7071 = 0.29$.