\ begin {reunir*} 1+ r +r^2 +\ cdots+ r^ {n-1} =\ frac {r^n-1} {r-1}\ qquad\ text {proporcionado $r\ ne 1$}\ end {reunir*}

It looks like you've provided a mathematical expression in LaTeX format. I'll rewrite it correctly and explain it.

### LaTeX Code:
```latex
\begin{align*}
1 + r + r^2 + \cdots + r^{n-1} = \frac{r^n - 1}{r - 1} \quad \text{provided that } r \neq 1
\end{align*}
```

### Explanation:
This formula represents the sum of a geometric series. Specifically, the series $1 + r + r^2 + \cdots + r^{n-1}$ sums the first $n$ terms of a geometric progression where the common ratio is $r$. The formula for the sum of the first $n$ terms of a geometric series is given by:

\[
S_n = \frac{r^n - 1}{r - 1}, \quad \text{for } r \neq 1
\]

Here:
- $ r $ is the common ratio.
- $ n $ is the number of terms.

The condition $ r \neq 1 $ is required because the formula involves division by $ r - 1 $, and if $ r = 1 $, this would result in division by zero, which is undefined.

### Key Points:
1. **Geometric Series**: A sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.
2. **Sum Formula**: The formula $ S_n = \frac{r^n - 1}{r - 1} $ is used to calculate the sum of the first $n$ terms of the geometric series.
3. **Condition $ r \neq 1 $**: This ensures that the formula is valid, as $ r = 1 $ would make the denominator zero.
4. **Special Case ( $ r = 1 $ )**: If $ r = 1 $, the series becomes $ 1 + 1 + 1 + \cdots + 1 $, and the sum would simply be $ n $.
5. **Applications**: Geometric series appear frequently in finance, computer science, physics, and other fields.

Would you like more details or have any questions about this? Here are some related questions you might find interesting:

1. How is the formula for the sum of an infinite geometric series derived?
2. Can you explain how to find the sum if $ r $ is a fraction (e.g., $ r = \frac{1}{2} $)?
3. What happens to the series sum when $ r > 1 $?
4. How would you apply this formula in a real-world scenario, such as calculating compound interest?
5. What is the sum of a geometric series if the number of terms $ n $ tends to infinity?

**Tip:** When dealing with geometric series, always check the value of the common ratio $ r $ to determine whether the series converges (sum approaches a finite limit) or diverges (sum grows without bound).

Explore the formula and application of the sum of a geometric series. This detailed explanation covers the series $1 + r + r^2 + \cdots + r^{n-1}$ with the formula $ S_n = \frac{r^n - 1}{r - 1} $, essential in mathematics and related fields.

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