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Sagot :
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]
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]
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