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To solve this problem, let’s go through each part step-by-step.

### Step 1: Identifying the Series
The series given is:
\[
\sum_{n=0}^{\infty} (-11)^n (x)^n
\]
We can rewrite this as:
\[
\sum_{n=0}^{\infty} \left( (-11x)^n \right)
\]

### Step 2: First Few Terms of the Series
To find the first few terms, substitute \( n = 0, 1, 2, \dots \):

1. For \( n = 0 \): \( (-11x)^0 = 1 \)
2. For \( n = 1 \): \( (-11x)^1 = -11x \)
3. For \( n = 2 \): \( (-11x)^2 = 121x^2 \)
4. For \( n = 3 \): \( (-11x)^3 = -1331x^3 \)

So, the first few terms of the series are:
\[
1 - 11x + 121x^2 - 1331x^3 + \dots
\]

### Step 3: Finding \( a \) and \( r \)
In a geometric series of the form:
\[
a + ar + ar^2 + \dots
\]
the first term \( a \) is the initial term of the series, and the common ratio \( r \) is the factor multiplied to each successive term.

From the given series, we have:
- The first term \( a = 1 \).
- The common ratio \( r = -11x \).

### Step 4: Sum of the Infinite Geometric Series
The sum \( S \) of an infinite geometric series with \( |r| < 1 \) is given by:
\[
S = \frac{a}{1 - r}
\]
Substitute \( a = 1 \) and \( r = -11x \):
\[
S = \frac{1}{1 - (-11x)} = \frac{1}{1 + 11x}
\]

### Step 5: Convergence Condition
For the series to converge, the absolute value of the common ratio \( r \) must be less than 1:
\[
|r| = |-11x| < 1
\]
This simplifies to:
\[
|x| < \frac{1}{11}
\]

### Final Answers
- \( a = 1 \)
- \( r = -11x \)
- The sum of the series, \( S = \frac{1}{1 + 11x} \), for \( |x| < \frac{1}{11} \).

Would you like further details on any step or have any questions?

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#### Related Questions
1. How do you determine the sum of a finite geometric series?
2. What is the significance of the convergence condition in infinite series?
3. How would the sum formula change if \( a \) or \( r \) were different?
4. What happens if \( x \) is outside the interval \( \left(-\frac{1}{11}, \frac{1}{11}\right) \)?
5. How do you apply the ratio test to verify the convergence of a geometric series?

#### Tip
For geometric series, always identify the first term \( a \) and the common ratio \( r \) first; these values determine both the sum and convergence conditions.

For the given geometric series, write out the first few terms of the series to find a and r, and find the sum of the series. Then express the inequality |r| < 1 in terms of x and find the values of x for which the inequality holds and the series converges.

Learn how to determine the sum of an infinite geometric series and express the convergence condition |r| < 1 in terms of x. This problem explores geometric series, finding first terms, the common ratio, and sum conditions. Suitable for students in Grades 10-12.

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