Calculate the number of DNA copies after 10 cycles.

In PCR, the number of DNA copies doubles with each cycle. The given series $2, 4, 8, 16, \dots$ indicates that after $n$ cycles, there will be $2^n$ copies of DNA.
For 10 cycles, the number of copies is:

$$\text{Copies} = 2^{10}$$ $$\text{Copies} = 1024$$

The number of DNA copies synthesized at the end of 10 cycles is $\boxed{\text{1024 copies}}$.

First, determine how many cycles can be completed in 30 minutes. Each cycle takes 2 minutes.

$$\text{Number of cycles} = \frac{\text{Total time}}{\text{Time per cycle}}$$ $$\text{Number of cycles} = \frac{30 \text{ minutes}}{2 \text{ minutes/cycle}}$$ $$\text{Number of cycles} = 15 \text{ cycles}$$

Now, calculate the number of DNA copies after 15 cycles:

$$\text{Copies} = 2^{15}$$ $$\text{Copies} = 32768$$

The number of DNA copies synthesized at the end of 30 minutes is $\boxed{\text{32768 copies}}$.