To find the sum of the given series:

$\sum_{n=0}^{\infty} 2 \left(\frac{5}{6}\right)^n$

Step 1: Identify the type of series

This is a geometric series of the form: $\sum_{n=0}^\infty ar^n$ where $a$ is the first term, and $r$ is the common ratio.

Step 2: Define the values

Here:

• $a = 2$ (the first term when $n = 0$)
• $r = \frac{5}{6}$ (the common ratio)

Step 3: Convergence of the series

A geometric series converges if $|r| < 1$. Here: $|r| = \left|\frac{5}{6}\right| = \frac{5}{6} < 1$ Since $|r| < 1$, the series converges.

Step 4: Sum of the series

The sum of an infinite geometric series is given by: $S = \frac{a}{1 - r}$ Substitute $a = 2$ and $r = \frac{5}{6}$: $S = \frac{2}{1 - \frac{5}{6}}$

Step 5: Simplify the denominator

$1 - \frac{5}{6} = \frac{6}{6} - \frac{5}{6} = \frac{1}{6}$

So: $S = \frac{2}{\frac{1}{6}} = 2 \cdot 6 = 12$

Final Answer:

The sum of the series is: $\boxed{12}$

Let me know if you'd like further clarifications or additional examples.

Here are some related questions to explore:

1. What happens if $r > 1$? Does the series converge?
2. Can you derive the sum formula for a finite geometric series?
3. How does the convergence test change if $n$ starts at 1 instead of 0?
4. What are the real-world applications of geometric series?
5. How do changes in $a$ or $r$ affect the sum of the series?

Tip: Always check if $|r| < 1$ before attempting to sum an infinite geometric series!