Let's analyze each part of the question in the image regarding infinite geometric series and determine whether each is convergent or divergent. If the series converges, we'll also compute the sum.

Problem 13:

Decide if each infinite geometric series is convergent or divergent. State the sum of the series if it exists.

a. $-64 + 16 - 4 + 1 - \dots$

• First term $a = -64$
• Common ratio $r = \frac{16}{-64} = -\frac{1}{4}$

For a geometric series to converge, the common ratio must satisfy $|r| < 1$. Here $r = -\frac{1}{4}$, which satisfies $|r| < 1$, so the series converges.

The sum $S$ of an infinite geometric series is given by: $S = \frac{a}{1 - r}$

Substituting the values: $S = \frac{-64}{1 - (-\frac{1}{4})} = \frac{-64}{1 + \frac{1}{4}} = \frac{-64}{\frac{5}{4}} = -64 \times \frac{4}{5} = -51.2$

So, the series converges, and the sum is -51.2.

b. $\frac{5}{12} - \frac{5}{6} + \frac{5}{3} - \frac{10}{3} + \dots$

This appears to be a geometric series where:

• First term $a = \frac{5}{12}$
• Common ratio $r = \frac{-\frac{5}{6}}{\frac{5}{12}} = -2$

Since the common ratio $r = -2$, which satisfies $|r| > 1$, this series diverges.

c. $6.1 + 1.22 + 0.244 + 0.0488 + \dots$

• First term $a = 6.1$
• Common ratio $r = \frac{1.22}{6.1} = 0.2$

Since $|r| < 1$, this series converges.

The sum $S$ is given by: $S = \frac{a}{1 - r}$ Substituting the values: $S = \frac{6.1}{1 - 0.2} = \frac{6.1}{0.8} = 7.625$

So, the series converges, and the sum is 7.625.

d. $\frac{24}{5} - 12 + 30 - 75 + \dots$

This is a geometric series where:

• First term $a = \frac{24}{5}$
• Common ratio $r = \frac{-12}{\frac{24}{5}} = -\frac{5}{2}$

Since $r = -\frac{5}{2}$ and $|r| > 1$, the series diverges.

Summary:

• a. Converges, sum = $-51.2$
• b. Diverges
• c. Converges, sum = $7.625$
• d. Diverges

Let me know if you need further explanation on any part!

Here are 5 related questions to consider:

1. How does the value of the common ratio affect convergence in a geometric series?
2. Can an infinite series with a common ratio greater than 1 ever converge?
3. How do you determine the common ratio from terms of a series?
4. What are some practical applications of geometric series in real-world scenarios?
5. How do we handle the sum of a finite geometric series?

Tip: When checking for convergence, always ensure that the absolute value of the common ratio $|r|$ is less than 1 for an infinite geometric series to converge.