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?

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.