Math / Calculus

Questionreach series will converge or diverge. Then sible. 7) $8.1+3.24+1.296+0.5184 \ldots$

Studdy Solution

STEP 1

1. The given series is an infinite geometric series.
2. To determine convergence or divergence, we need to identify the common ratio.
3. A geometric series converges if the absolute value of the common ratio is less than 1.

STEP 2

1. Identify the first term and the common ratio of the series.
2. Determine if the series converges or diverges based on the common ratio.
3. If the series converges, calculate the sum of the series.

STEP 3

Identify the first term and the common ratio of the series:
The first term $a$ is $8.1$ .
To find the common ratio $r$ , divide the second term by the first term:
$r = \frac{3.24}{8.1}$
Calculate the ratio:
$r = \frac{3.24}{8.1} = 0.4$

STEP 4

Determine if the series converges or diverges:
A geometric series converges if $|r| < 1$ .
Here, $|r| = |0.4| = 0.4$ , which is less than 1.
Therefore, the series converges.

STEP 5

Calculate the sum of the series, since it converges:
The sum $S$ of an infinite geometric series is given by:
$S = \frac{a}{1 - r}$
Substitute the values of $a$ and $r$ :
$S = \frac{8.1}{1 - 0.4} = \frac{8.1}{0.6}$
Calculate the sum:
$S = 13.5$
The series converges, and the sum is $\boxed{13.5}$ .

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Questionreach series will converge or diverge. Then sible. 7) $8.1+3.24+1.296+0.5184 \ldots$

Studdy Solution

STEP 1

1. The given series is an infinite geometric series.
2. To determine convergence or divergence, we need to identify the common ratio.
3. A geometric series converges if the absolute value of the common ratio is less than 1.

STEP 2

1. Identify the first term and the common ratio of the series.
2. Determine if the series converges or diverges based on the common ratio.
3. If the series converges, calculate the sum of the series.

STEP 3

Identify the first term and the common ratio of the series:
The first term $a$ is $8.1$ .
To find the common ratio $r$ , divide the second term by the first term:
$r = \frac{3.24}{8.1}$
Calculate the ratio:
$r = \frac{3.24}{8.1} = 0.4$

STEP 4

Determine if the series converges or diverges:
A geometric series converges if $|r| < 1$ .
Here, $|r| = |0.4| = 0.4$ , which is less than 1.
Therefore, the series converges.

STEP 5

Calculate the sum of the series, since it converges:
The sum $S$ of an infinite geometric series is given by:
$S = \frac{a}{1 - r}$
Substitute the values of $a$ and $r$ :
$S = \frac{8.1}{1 - 0.4} = \frac{8.1}{0.6}$
Calculate the sum:
$S = 13.5$
The series converges, and the sum is $\boxed{13.5}$ .

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Questionreach series will converge or diverge. Then sible. 7) 8.1+3.24+1.296+0.5184…8.1+3.24+1.296+0.5184 \ldots8.1+3.24+1.296+0.5184…

Studdy Solution

STEP 1

1. The given series is an infinite geometric series.2. To determine convergence or divergence, we need to identify the common ratio.3. A geometric series converges if the absolute value of the common ratio is less than 1.

STEP 2

1. Identify the first term and the common ratio of the series.2. Determine if the series converges or diverges based on the common ratio.3. If the series converges, calculate the sum of the series.

STEP 3

Identify the first term and the common ratio of the series:The first term a a a is 8.1 8.1 8.1.To find the common ratio r r r, divide the second term by the first term:r=3.248.1 r = \frac{3.24}{8.1} r=8.13.24​Calculate the ratio:r=3.248.1=0.4 r = \frac{3.24}{8.1} = 0.4 r=8.13.24​=0.4

STEP 4

Determine if the series converges or diverges:A geometric series converges if ∣r∣<1 |r| < 1 ∣r∣<1.Here, ∣r∣=∣0.4∣=0.4 |r| = |0.4| = 0.4 ∣r∣=∣0.4∣=0.4, which is less than 1.Therefore, the series converges.

STEP 5

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Questionreach series will converge or diverge. Then sible. 7) $8.1+3.24+1.296+0.5184 \ldots$

1. The given series is an infinite geometric series.
2. To determine convergence or divergence, we need to identify the common ratio.
3. A geometric series converges if the absolute value of the common ratio is less than 1.

1. Identify the first term and the common ratio of the series.
2. Determine if the series converges or diverges based on the common ratio.
3. If the series converges, calculate the sum of the series.

Identify the first term and the common ratio of the series:
The first term $a$ is $8.1$ .
To find the common ratio $r$ , divide the second term by the first term:
$r = \frac{3.24}{8.1}$
Calculate the ratio:
$r = \frac{3.24}{8.1} = 0.4$

Determine if the series converges or diverges:
A geometric series converges if $|r| < 1$ .
Here, $|r| = |0.4| = 0.4$ , which is less than 1.
Therefore, the series converges.