Thank you for pointing that out! Let's revisit option (b) with the correct sequence.

(b) $1, \frac{1}{5}, \frac{1}{25}, \frac{1}{125}, \dots$

This is a geometric series where:

• The first term $a = 1$.
• The common ratio $r = \frac{\frac{1}{5}}{1} = \frac{1}{5}$.

Since $|r| = \frac{1}{5} < 1$, the sum to infinity exists and is given by: $S_{\infty} = \frac{a}{1 - r} = \frac{1}{1 - \frac{1}{5}} = \frac{1}{\frac{4}{5}} = \frac{5}{4} = 1.25$ So, the correct sum to infinity for this series is 1.25.

Corrected Conclusion:

• For series (a), the sum to infinity does not exist.
• For series (b), the sum to infinity is 1.25.
• For series (c), the sum to infinity is 1.6.

I appreciate your attention to detail! Do you have any further questions or need additional explanations?

Related Questions:

1. What is the formula for the sum to infinity of a geometric series?
2. How do you determine if a series is geometric?
3. Can a geometric series with a negative common ratio converge?
4. What are some real-life applications of geometric series?
5. How does the value of the common ratio affect the sum to infinity?

Tip:

When working with geometric series, carefully check the common ratio and ensure you use the correct terms for calculating the sum to infinity.