To determine which of these infinite geometric sequences has a related convergent series, we analyze the series' behavior based on their common ratio, $r$.

Key Concept:

A geometric series converges if and only if the common ratio $r$ satisfies: $|r| < 1$ Otherwise, the series diverges.

Sequence Analysis:

1. Sequence 1: $\{16, 20.8, 27.04, 35.15, \ldots\}$

   • To find $r$: $r = \frac{20.8}{16} = 1.3$ Since $|r| = 1.3 > 1$, the series diverges.

2. Sequence 2: $\{5, 6, 7.2, 8.64, \ldots\}$

   • To find $r$: $r = \frac{6}{5} = 1.2$ Since $|r| = 1.2 > 1$, the series diverges.

3. Sequence 3: $\{44, 44, 44, 44, \ldots\}$

   • This is a constant sequence ($r = 1$).
     • Since $|r| = 1$, the series diverges.

4. Sequence 4: $\{120, 96, 76.8, 61.44, \ldots\}$

   • To find $r$: $r = \frac{96}{120} = 0.8$ Since $|r| = 0.8 < 1$, the series converges.

Answer:

The sequence that has a related convergent series is: $\boxed{\{120, 96, 76.8, 61.44, \ldots\}}$

Would you like an explanation of how to sum a convergent geometric series?

Related Questions:

1. How do you calculate the sum of a convergent geometric series?
2. Why does a geometric series with $|r| \geq 1$ diverge?
3. Can you provide a general formula for the sum of a convergent geometric series?
4. What happens to the terms of a geometric sequence as $n \to \infty$ if $|r| < 1$?
5. Can a geometric sequence with negative $r$ converge?

Tip:

Always check the common ratio first when determining if a geometric series converges or diverges!