In the context of a geometric series, $r$ is called the common ratio. It represents the factor by which each term in the series is multiplied to get the next term.

For the series $\sum_{k=1}^{n} 2^k$, each term is of the form $2^k$. The common ratio $r$ is the factor that you multiply one term by to get the next term.

Given that the terms in the series are $2^1, 2^2, 2^3, \dots, 2^n$, you can see that:

$\frac{2^2}{2^1} = \frac{2^3}{2^2} = \frac{2^4}{2^3} = \dots = 2$

Thus, in this series, the common ratio $r = 2$.

Would you like further clarification, or do you have any more questions?

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Related Questions:

1. How do you find the common ratio in a geometric series?
2. What is the difference between an arithmetic and a geometric series?
3. Can the common ratio $r$ be a negative number?
4. How does the value of $r$ affect the sum of a geometric series?
5. What is the common ratio in the series $\sum_{k=1}^{n} 3^k$?

Tip: The common ratio $r$ is crucial in determining the behavior of a geometric series, especially when calculating its sum or analyzing its convergence.