1. Let's start by stating the problem: solving a series means finding the sum of its terms, either finite or infinite. 2. The general formula depends on the type of series. For example, for an arithmetic series, the sum of the first $n$ terms is given by: $$S_n = \frac{n}{2} (a_1 + a_n)$$ where $a_1$ is the first term and $a_n$ is the $n$th term. 3. For a geometric series, the sum of the first $n$ terms is: $$S_n = a_1 \frac{1 - r^n}{1 - r}$$ where $r$ is the common ratio and $r \neq 1$. 4. Important rules: - Identify the type of series (arithmetic, geometric, or other). - Use the appropriate formula. - For infinite geometric series with $|r| < 1$, the sum is: $$S = \frac{a_1}{1 - r}$$ 5. Example: Solve the sum of the series $2 + 4 + 8 + 16 + ... + 2^n$. 6. This is a geometric series with $a_1 = 2$ and $r = 2$. 7. Using the formula for the sum of the first $n$ terms: $$S_n = 2 \frac{1 - 2^n}{1 - 2}$$ 8. Simplify the denominator: $$1 - 2 = -1$$ 9. So, $$S_n = 2 \frac{1 - 2^n}{-1} = -2 (1 - 2^n)$$ 10. Distribute the negative sign: $$S_n = -2 + 2 \cdot 2^n = -2 + 2^{n+1}$$ 11. Final answer: $$S_n = 2^{n+1} - 2$$ This is how you solve a geometric series sum.