The sum of frac{1}{2} + frac{1}{3} cdot frac{ | vedclass

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(B) The given series is $S = \frac{1}{2} + \frac{1}{3} \cdot \frac{1}{2^3} + \frac{1}{5} \cdot \frac{1}{2^5} + \dots \infty$. We know the logarithmic

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$\log_e \sqrt{3}$

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(B) The given series is $S = \frac{1}{2} + \frac{1}{3} \cdot \frac{1}{2^3} + \frac{1}{5} \cdot \frac{1}{2^5} + \dots \infty$. We know the logarithmic series expansion: $\log_e \left( \frac{1+x}{1-x} \right) = 2 \left( x + \frac{x^3}{3} + \frac{x^5}{5} + \dots \infty \right)$ for $|x| Let $x = \frac{1}{2}$. Then,$\log_e \left( \frac{1 + 1/2}{1 - 1/2} \right) = 2 \left( \frac{1}{2} + \frac{1}{3} \cdot \frac{1}{2^3} + \frac{1}{5} \cdot \frac{1}{2^5} + \dots \infty \right)$. $\log_e \left( \frac{3/2}{1/2} \right) = 2S$. $\log_e(3) = 2S$. $S = \frac{1}{2} \log_e(3) = \log_e(3^{1/2}) = \log_e \sqrt{3}$.

The sum of $\frac{1}{2} + \frac{1}{3} \cdot \frac{1}{2^3} + \frac{1}{5} \cdot \frac{1}{2^5} + \dots \infty$ is

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