The expression given is:

$\sum_{n=1}^{\infty} \left( \left( \frac{1}{2} \right)^n + \left( \frac{1}{3} \right)^n \right)$

To evaluate this sum, we can break it down into two separate series:

$\sum_{n=1}^{\infty} \left( \frac{1}{2} \right)^n + \sum_{n=1}^{\infty} \left( \frac{1}{3} \right)^n$

Each of these series is a geometric series, and we can solve each separately.

Step 1: Sum of the first series

The first series is:

$\sum_{n=1}^{\infty} \left( \frac{1}{2} \right)^n$

This is a geometric series with a first term $a = \frac{1}{2}$ and a common ratio $r = \frac{1}{2}$. The sum of an infinite geometric series starting from $n=1$ is given by:

$S = \frac{a}{1 - r}$

For this series:

$S_1 = \frac{\frac{1}{2}}{1 - \frac{1}{2}} = \frac{\frac{1}{2}}{\frac{1}{2}} = 1$

Step 2: Sum of the second series

The second series is:

$\sum_{n=1}^{\infty} \left( \frac{1}{3} \right)^n$

This is also a geometric series with a first term $a = \frac{1}{3}$ and a common ratio $r = \frac{1}{3}$. Applying the formula for the sum of an infinite geometric series:

$S_2 = \frac{\frac{1}{3}}{1 - \frac{1}{3}} = \frac{\frac{1}{3}}{\frac{2}{3}} = \frac{1}{2}$

Final Calculation

Adding the results from both series:

$S = S_1 + S_2 = 1 + \frac{1}{2} = \frac{3}{2}$

Answer

The value of the given expression is:

$\frac{3}{2}$

Would you like more details on any of the steps?

Here are some additional related questions:

1. How is the sum formula for a geometric series derived?
2. What happens if the common ratio in a geometric series is greater than 1?
3. How do geometric series apply in real-world contexts?
4. Can you find a sum if the series starts from $n=0$ instead of $n=1$?
5. What changes if we consider finite geometric series instead of infinite ones?

Tip: Geometric series converge only when the common ratio $r$ satisfies $|r| < 1$.