Infinite array of capacitors and inductors You may be familiar with the surprising result one gets when calculating the equivalent of an infinite array of resistors. What if we change this circuit and replace the resistors with capacitors and inductors?
Following the notation given in the link I've provided above, let's replace $R_1$ with $C$ and $R_2$ with $L$ so that
$$R_1 \rightarrow Z_1=\frac{-j}{\omega C}$$
$$R_2 \rightarrow Z_2=j\omega L$$
where $j$ is the imaginary unit. Again using the result written in that link, we get the following equation:
$$Z_{eq}^2-\frac{j}{\omega C}Z_{eq}-\frac{L}{C}=0$$
Solving this quadratic equation we get that:
$$Z_{eq}=\frac{\frac{j}{\omega C}\pm\sqrt{\frac{-1}{\omega^2C^2}+4\frac{L}{C}}}{2}=\pm\sqrt{\frac{L}{C}-\frac{1}{(2\omega C)^2}}+j\frac{1}{2\omega C}$$
So an array of ideal capacitors and inductors lead to a complex (not imaginary) equivalent impedance if $L>\frac{1}{4C\omega^2}$. This means that, if the circuit was fed with a source, actual power would be dissipated, even though each of the individual impedances are purely reactive. How does this make sense?
 A: Initially I thought you had just rediscovered the Telegrapher's Equations - but then I realized you had your capacitors and inductors "the other way around" from that more usual scenario (described here)
Even though your situation is unusual, there is a way to understand what is happening. The capacitors in your network are charging up - and while some of that charging is transient, some of it "goes on forever" because of the infinite extent of the network. This charging of the capacitors means there is a mechanism for storing energy - and I think that's what your equations are telling you.
This is much easier to understand when you switch your capacitors (to be $R_2$ in your diagram) and inductors (to be $R_1$). When you do that, you end up with an expression for the impedance that will tend to $Z=\sqrt{\frac{L}{C}}$ when you make $L$ and $C$ infinitesimal while preserving their ratio (which is what happens when you consider a transmission line as being made up of many small inductors and capacitors).
When you have an ordinary transmission line, a pulse will propagate, and energy will be stored per unit length. The storing of energy is indistinguishable from dissipation of energy (until you get a reflection or some other mechanism to extract the energy again).
