frac{1}{5} + frac{1}{2} cdot frac{1}{5^2} + f | vedclass

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Question

(C) The given series is of the form $\frac{x}{1} + \frac{x^2}{2} + \frac{x^3}{3} + \dots = -\ln(1-x)$,where $x = \frac{1}{5}$.Substituting $x = \frac{

Vedclass · Accepted Answer

$2{\log _e} \frac{\sqrt{5}}{2}$

Vedclass · Answer

(C) The given series is of the form $\frac{x}{1} + \frac{x^2}{2} + \frac{x^3}{3} + \dots = -\ln(1-x)$,where $x = \frac{1}{5}$.Substituting $x = \frac{1}{5}$ into the series expansion:$S = -\ln(1 - \frac{1}{5}) = -\ln(\frac{4}{5}) = \ln(\frac{5}{4})$.We can rewrite $\frac{5}{4}$ as $(\frac{\sqrt{5}}{2})^2$.Therefore,$S = \ln((\frac{\sqrt{5}}{2})^2) = 2\ln(\frac{\sqrt{5}}{2})$.Thus,the correct option is $C$.

$\frac{1}{5} + \frac{1}{2} \cdot \frac{1}{5^2} + \frac{1}{3} \cdot \frac{1}{5^3} + \dots \infty = $

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