Math / Calculus

Question23-32 Determine whether the geometric series is convergent or divergent. If it is convergent, find its sum.
25. $10-2+0.4-0.08+\ldots$

Studdy Solution

STEP 1

1. The given series is a geometric series.
2. A geometric series is of the form $a + ar + ar^2 + ar^3 + \ldots$ .
3. A geometric series converges if the absolute value of the common ratio $|r| < 1$ .
4. If the series converges, its sum can be calculated using the formula $S = \frac{a}{1 - r}$ .

STEP 2

1. Identify the first term and the common ratio of the series.
2. Determine if the series is convergent or divergent.
3. If convergent, calculate the sum of the series.

STEP 3

Identify the first term ( $a$ ) and the common ratio ( $r$ ) of the series:
The given series is $10 - 2 + 0.4 - 0.08 + \ldots$ .
First term, $a = 10$ .
To find the common ratio $r$ , divide the second term by the first term:
$r = \frac{-2}{10} = -0.2$

STEP 4

Determine if the series is convergent or divergent:
A geometric series converges if the absolute value of the common ratio $|r| < 1$ .
Calculate $|r|$ :
$|r| = |-0.2| = 0.2$
Since $0.2 < 1$ , the series is convergent.

STEP 5

Calculate the sum of the series since it is convergent:
Use the formula for the sum of a convergent geometric series:
$S = \frac{a}{1 - r}$
Substitute the values of $a$ and $r$ :
$S = \frac{10}{1 - (-0.2)} = \frac{10}{1 + 0.2} = \frac{10}{1.2}$
Simplify the fraction:
$S = \frac{10}{1.2} = \frac{100}{12} = \frac{25}{3}$
The series is convergent, and its sum is $\boxed{\frac{25}{3}}$ .

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