Why do we generally add capacitors in series with AC sources when DC also exists in the circuit?
We don't normally place capacitors in front of AC sources.
What you are calling an AC source in your question is actually an &quot;AC signal source&quot;, (as opposed to an AC power source). In that case, we might place what is called a DC-blocking capacitor in series with the AC source which allows the AC source to &quot;ride on top of&quot; or superimpose itself onto the DC-bias. Along with a resistor in series with the DC source, it protects the sources from each other, and prevents the sources from fighting each other where both sources meet which allows both to appear together on the line. Without them, it would be unstoppable force vs immovable object for ideal sources.
In your example, a resistor in series with the AC source present a voltage drop which distorts the voltage signal which you do not need to live with because a properly chosen capacitor won't do this.
Why do we use a resistor in series with the DC source and not an inductor (for AC-blocking)? Because if you try and use the capacitors and inductor together they will have resonance effects. Real inductors also do not approach the ideal as closely as resistors or capacitors so we don't try to get rid of the cap and just use an inductor alone in series with the DC source and and resistor in series with the AC source.