frac{1}{2} - frac{1}{2 cdot 2^2} + frac{1}{3 | vedclass

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(C) The given series is $\frac{1}{2} - \frac{1}{2 \cdot 2^2} + \frac{1}{3 \cdot 2^3} - \frac{1}{4 \cdot 2^4} + \ldots$ We know the logarithmic expansi

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$\log _e\left(\frac{3}{2}\right)$

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(C) The given series is $\frac{1}{2} - \frac{1}{2 \cdot 2^2} + \frac{1}{3 \cdot 2^3} - \frac{1}{4 \cdot 2^4} + \ldots$ We know the logarithmic expansion: $\log _e(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots$ Comparing the given series with the expansion,we set $x = \frac{1}{2}$. Substituting $x = \frac{1}{2}$ into the expansion,we get: $\log _e\left(1 + \frac{1}{2}\right) = \frac{1}{2} - \frac{(1/2)^2}{2} + \frac{(1/2)^3}{3} - \frac{(1/2)^4}{4} + \ldots$ $= \log _e\left(\frac{3}{2}\right)$.

$\frac{1}{2} - \frac{1}{2 \cdot 2^2} + \frac{1}{3 \cdot 2^3} - \frac{1}{4 \cdot 2^4} + \ldots$ is equal to

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