To determine which of the given geometric series converges, we need to analyze each series. A geometric series converges if and only if the absolute value of its common ratio [tex]\( r \)[/tex] is less than 1 ([tex]\(|r| < 1\)[/tex]). Now, let's evaluate each series:
1. First Series:
[tex]\[
\frac{1}{81} + \frac{1}{27} + \frac{1}{9} + \frac{1}{3} + \ldots
\][/tex]
- Identify the first term [tex]\( a \)[/tex]: [tex]\( a = \frac{1}{81} \)[/tex]
- Find the common ratio [tex]\( r \)[/tex]:
[tex]\[
r = \frac{\frac{1}{27}}{\frac{1}{81}} = 3
\][/tex]
- Since [tex]\( |r| = 3 \)[/tex], and [tex]\( 3 > 1 \)[/tex], this series does not converge.
2. Second Series:
[tex]\[
1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots
\][/tex]
- Identify the first term [tex]\( a \)[/tex]: [tex]\( a = 1 \)[/tex]
- Find the common ratio [tex]\( r \)[/tex]:
[tex]\[
r = \frac{\frac{1}{2}}{1} = \frac{1}{2}
\][/tex]
- Since [tex]\( |r| = \frac{1}{2} \)[/tex], and [tex]\( \frac{1}{2} < 1 \)[/tex], this series converges.
3. Third Series:
[tex]\[
\sum_{n=1}^{\infty} 7(-4)^{n-1}
\][/tex]
- This is a geometric series with:
[tex]\[
a = 7 \quad \text{and} \quad r = -4
\][/tex]
- Since [tex]\( |r| = 4 \)[/tex], and [tex]\( 4 > 1 \)[/tex], this series does not converge.
4. Fourth Series:
[tex]\[
\sum_{n=1}^{\infty} \frac{1}{5} \cdot 2^{n-1}
\][/tex]
- This is a geometric series with:
[tex]\[
a = \frac{1}{5} \quad \text{and} \quad r = 2
\][/tex]
- Since [tex]\( |r| = 2 \)[/tex], and [tex]\( 2 > 1 \)[/tex], this series does not converge.
Among the given series, only the second series converges:
[tex]\[
1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \ldots
\][/tex]
