Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.) n = 1 3(0.4)n-1

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Chapter1: Functions And Models
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**Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)**

\[
\sum_{n=1}^{\infty} 3(0.4)^{n-1}
\]

**Explanation:**

The given series is a geometric series. In general, a geometric series has the form \(\sum_{n=1}^{\infty} ar^{n-1}\), where:
- \(a\) is the first term of the series.
- \(r\) is the common ratio.

In this series:
- The first term \(a = 3\).
- The common ratio \(r = 0.4\).

To determine whether the series is convergent, check the absolute value of the common ratio \( |r| < 1 \). If true, the series converges. If not, it diverges.

The sum \(S\) of a convergent geometric series can be found using the formula:
\[ S = \frac{a}{1 - r} \]

In this case, since \( |0.4| < 1 \), the series converges. You can calculate the sum using:
\[ S = \frac{3}{1 - 0.4} \]

Substituting the values:
\[ S = \frac{3}{0.6} = 5 \]

Thus, the series is convergent, and its sum is 5.
Transcribed Image Text:**Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)** \[ \sum_{n=1}^{\infty} 3(0.4)^{n-1} \] **Explanation:** The given series is a geometric series. In general, a geometric series has the form \(\sum_{n=1}^{\infty} ar^{n-1}\), where: - \(a\) is the first term of the series. - \(r\) is the common ratio. In this series: - The first term \(a = 3\). - The common ratio \(r = 0.4\). To determine whether the series is convergent, check the absolute value of the common ratio \( |r| < 1 \). If true, the series converges. If not, it diverges. The sum \(S\) of a convergent geometric series can be found using the formula: \[ S = \frac{a}{1 - r} \] In this case, since \( |0.4| < 1 \), the series converges. You can calculate the sum using: \[ S = \frac{3}{1 - 0.4} \] Substituting the values: \[ S = \frac{3}{0.6} = 5 \] Thus, the series is convergent, and its sum is 5.
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