Equivalent capacitance of an infinite system

[Fahamedi]

Homework Statement

Find the equivalent capacitance of the infinite system between points a and b (see figure).
Where Vi is the potential difference in the number i capacitor.

Homework Equations

Q=CV

The Attempt at a Solution

For number i capacitors we have the relations

[itex] Q_i=CV_i[/itex] and [itex]Q'_i=CV'_i[/itex]
[itex]\Rightarrow[/itex] [itex]Q\equiv[/itex] [itex]\sum_{i=0}^{\infty}[/itex][itex](Q_i+Q'_i)[/itex]=C[itex]\sum_{i=0}^{\infty}[/itex][itex](Vi+V'i)[/itex]

Now if V is the potential difference V=Va-Vb and we look at the paths in the system,

[itex] V_0+V'_0=V[/itex] ; [itex]V_0+V_1+V'_1=V[/itex] ; ... ; [itex]V_0+V_1+V_2+...+V_i+V'_i=V[/itex] ; ...

From this I found

[itex] V'_i=V_{i+1}+V'_{i+1}[/itex]

[itex]\Rightarrow[/itex] [itex]Q[/itex]=C[itex]\sum_{i=0}^{\infty}[/itex][itex](V_i+V_{i+1}+V'_{i+1})[/itex]

And I'm stuck here, I don't know what to do next, or should I do something different?

 

[vela]

Hint: Say you divide the circuit into two along the dotted line. What's the equivalent capacitance of the righthand portion?