The given series is a geometric series:
$2 + 4 + 8 + 16 + 32 + \dots$

In a geometric series, each term is multiplied by a constant ratio to get the next term. Here, the first term $a = 2$ and the common ratio $r = \frac{4}{2} = 2$.

If the series is infinite, the sum of an infinite geometric series is given by the formula: $S = \frac{a}{1 - r} \quad \text{where} \ |r| < 1.$ However, since $r = 2$, which is greater than 1, the series does not converge to a finite sum. Therefore, the sum of this infinite geometric series does not exist.

If the series is finite (e.g., stopping after a certain number of terms), the sum of the first $n$ terms of a geometric series is: $S_n = a \frac{r^n - 1}{r - 1}.$ For your series, with $a = 2$ and $r = 2$, the sum of the first $n$ terms is: $S_n = 2 \frac{2^n - 1}{2 - 1} = 2(2^n - 1).$

Do you want to calculate the sum for a specific number of terms, or further details on how this works?

Related questions:

1. What is the formula for the sum of an infinite geometric series with $|r| < 1$?
2. How does the common ratio affect the convergence of a geometric series?
3. Can a divergent series be used in real-world applications?
4. How do you derive the formula for the sum of a finite geometric series?
5. What happens if the common ratio is negative in a geometric series?

Tip: For any geometric series, always check the common ratio to determine if the series converges or diverges.