If the following infinite geometric series converges, find its sum.
\[\large\displaystyle\sum\limits_{k=0}^{{\infty}}{{{2(3)^{k}}}} = \]

To determine if the infinite geometric series converges and to find its sum, we first need to identify the first term \(a\) and the common ratio \(r\) of the series.

The given series is:

\[
\sum\limits_{k=0}^{\infty} 2(3)^k
\]

This can be expressed as:

\[
2 + 2(3) + 2(3^2) + 2(3^3) + \dots
\]

### Identifying \(a\) and \(r\)
- The first term \(a\) of the series is the value of the expression when \(k = 0\):
  \[
  a = 2(3)^0 = 2
  \]
- The common ratio \(r\) is the factor by which each term is multiplied to get the next term:
  \[
  r = 3
  \]

### Convergence Condition
For an infinite geometric series to converge, the common ratio \(r\) must satisfy:
\[
|r| < 1
\]
Here, \(r = 3\), and since \( |3| > 1 \), the series does not converge.

### Conclusion
Since the common ratio \(r = 3\) does not satisfy the convergence condition \( |r| < 1 \), the given infinite geometric series **diverges** and therefore does not have a sum.

Would you like more details or have any questions? Here are some related questions to explore:

1. What happens when the common ratio \(r\) is negative?
2. How do you find the sum of a convergent geometric series?
3. Can you provide an example of a convergent geometric series?
4. How do you calculate the sum of a finite geometric series?
5. What is the formula for the sum of an infinite geometric series?

**Tip:** In any geometric series, always check the common ratio \(r\) first to determine whether the series converges or diverges.

Explore the convergence of infinite geometric series using examples. Learn about the criteria for convergence and divergence, illustrated with detailed explanations and examples. Suitable for high school and above.

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