Parallel RLC circuit, how branching currents may each be larger than source current at resonance? How can the branching currents individually be greater than the source current?
 A: You might already know this, but in an ac circuit, currents in all the devices don't flow in the same phase. For example, if you connect an inductor to an ac source, current will flow in the inductor only after the ac source has finished a quarter cycle. 
Due to this, it is possible at some moments that a branch has more current magnitude than the source itself.

In the above picture, if you notice the dotted part ,the capacitor has more current than the source.
However if you vector add their phasors, you will always get total current equal to source current.
If this does not make sense to you, you can think of it this way: capacitors and inductors both have storage mechanisms, they both can store energy in the form of an electric or magnetic field and then use it later. Hence it is possible that at some point their branch has more current than the source.
However resistors can't do that. They dissipate energy all the time. Hence current in resistance branch has to be less than source current.
A: Total current is the vector sum of all currents
$$I_T=I_R + (I_L + I_C)$$
$I_L$ and $I_C$ are $180$ degrees out of phase at resonance, so the total current becomes
$I_T=I_R $
$$f_R=\frac{1}{2\pi\sqrt{LC}}$$
$$I_L=\frac{V}{2\pi f_R L}$$
$I_C=V.2\pi f_R C$
$$I_L=V\sqrt{\frac{C}{L}}$$
and $I_C=V\sqrt{\frac{ C}{L}}$
So, depending on the values of $L$ and $C$,  $I_L$ or $I_C$ can be greater than the total current $I_T$
