2 + 4 + 8 + 16 +32+... what is the sum of the geometric series

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.

2 + 4 + 8 + 16 + 32 + ... what is the sum of the geometric series

Discover how to find the sum of a geometric series like 2 + 4 + 8 + 16 + 32. Learn about the formulas for both finite and infinite geometric series, the concept of common ratios, and the conditions for series convergence. Suitable for high school students in Grades 9-12.

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