Determine whether the geometric series is convergent or divergent.10−4+1.6−0.64+….If it's convergent find its sum.

Step 1 Let S=10−4+1.60.64+… Given that S is a geometric series, The known fact is that a geometric series a+ar+ar2+… is convergent ⇔|r|&lt;1. By comparing the given series with its general form, a=10, r=−410=−0.4. |r|=|−0.4|&lt;1. This implies that the series 10−4+1.6−0.64+… is convergent. Step 2 The known fact is that a geometric series a+ar+ar2+… - is equal to a1−r(r&lt;1). 10−4+1.6−0.64±…=(10+1.6+0.256+…)−(4+0.64+…) =(101−0.16)−(41−0.16)[∵a=10,r=0.16 in first geometric seriesa=4,r=0,16in second geometric series] =(10−41−0.16) =(60.84) =7.1428571428571429

Question: "Determine whether the geometric series is convergent or..."

Priscilla Johnston

Answered question

2021-12-10

Determine whether the geometric series is convergent or divergent.

10 - 4 + 1.6 - 0.64 + \dots .

If it's convergent find its sum.

Answer & Explanation

Juan Spiller

Beginner2021-12-11Added 38 answers

Step 1
Let $S = 10 - 4 + 1.6 0.64 + \dots$
Given that S is a geometric series,
The known fact is that a geometric series $a + a r + a r^{2} + \dots$ is convergent $\Leftrightarrow | r | < 1$ .
By comparing the given series with its general form,
$a = 10$ , $r = \frac{- 4}{10} = - 0.4$ .
$| r | = | - 0.4 | < 1$ .
This implies that the series $10 - 4 + 1.6 - 0.64 + \dots$ is convergent.
Step 2
The known fact is that a geometric series $a + a r + a r^{2} + \dots$ - is equal to $\frac{a}{1 - r} (r < 1)$ .
$10 - 4 + 1.6 - 0.64 \pm \dots = (10 + 1.6 + 0.256 + \dots) - (4 + 0.64 + \dots)$
$= (\frac{10}{1 - 0.16}) - (\frac{4}{1 - 0.16}) [\begin{matrix} ∵ a = 10, r = 0.16 i n f i r s t g e o m e t r i c s e r i e s \\ a = 4, r = 0, 16 i n s e c o n d g e o m e t r i c s e r i e s \end{matrix}]$
$= (\frac{10 - 4}{1 - 0.16})$
$= (\frac{6}{0.84})$
$= 7.1428571428571429$

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