An infinite number of identical capacitors each of capacitance $1\,\mu F$ are connected as in adjoining figure. Then the equivalent capacitance between $A$ and $B$ is......$\mu F$
AIIMS 2018, Medium
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(b)This combination forms a $G.P.$ $S = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8}......$
Sum of infinite $G.P.$ $S = \frac{a}{{1 - r}}$
Here $a$ $=$ first term $= 1$ and $r$ $=$ common ratio $ = \frac{1}{2}$
$==>$ $S = \frac{1}{{1 - \frac{1}{2}}} = 2$ $==>$ ${C_{eq}} = 2\,\mu F$
Sum of infinite $G.P.$ $S = \frac{a}{{1 - r}}$
Here $a$ $=$ first term $= 1$ and $r$ $=$ common ratio $ = \frac{1}{2}$
$==>$ $S = \frac{1}{{1 - \frac{1}{2}}} = 2$ $==>$ ${C_{eq}} = 2\,\mu F$
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