Find the equivalent capacitance between `A` and `B`.

Explanation

(a) `(1)/(C_(eq))=(1)/(c)+(1)/(2C)+(1)/(4C)+…oo`
`=(1)/(c)(1+(1)/(2)+(1)/(2^(2))+…oo)`
`(1)/(c)((1)/(1-(1)/(2)))=(2)/(c)`
`C_(eq)=(c)/(2)`
(b) Equivalent capacitance:
In second branch, `(c)/(2)`, third branch `(c)/(4)`,...
`C,(c)/(2),(c)/(4),...oo` are in parallel
`C_(eq)=C+(c)/(2)+(c)/(4)+...oo`
`=C(1+(1)/(2)+(1)/(2^(2))+...oo)`
`=C((1)/(1-(1)/(2)))=2C`
(c) There are infinite pairs of `C` and `2C`. let equivalent capacitance be `C'`. Equivalent capacitance is same including/excluding on pair. (if you take one bucket of water from sea, there is no effect on volume of water in sea.)
`C` and `(2C+C')` are in series.
`C_(eq)=C'=(C(2C+C'))/(C+(2C+C'))`
`3C C'+C'^(2)=2c^(2)+C C'`
`C'^(2)+2C C'-2C^(2)=0`
`C'=-2C+-sqrt(4C^(2)+8C^(2))/(2)=-C+-sqrt(3)C`
`C'=(sqrt(3)-1)C`
Let `C`' be equivalent capacitance.
`(1)/(C_(eq))=(1)/(c)+(1)/(c)+(1)/((2C+C'))`
`=((2C+C')xx2+C)/(C(2C+C'))`
`(1)/(C')=(5C+2C')/(2C^(2)+C C')`
`2C^(2)+C C'=5C C'+2C'^(2)`
`2C'^(2)+4C C'-2C^(2)=0`
`C'^(2)+2C C'-C^(2)=0`
`C'=(-2C+-sqrt(4C^(2)+4C^(2)))/(2)=(-2C+-2sqrt(2)C)/(2)`
`=(sqrt(2)-1)C`
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