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.