Question

Determine whether the geometric series is convergent or divergent. 10 − 6 + 3.6 − 2.16 + convergent divergent Correct: Your answer is correct. If it is convergent, find its sum. (If the quantity diverges, enter DIVERGES.)

Answer 1

Answer:

The series CONVERGES with the sum being 6.25.

Step-by-step explanation:

Given that the geometric series is 10 - 6 + 3.6 - 2.16 + ....

So the first term of the geometric progression is, $a_{1}$ = 10

The second term of the geometric progression is $a_2$ = -6 = $a_1$ × r = 10 × r

where r is the constant ratio.

Therefore r  = $\frac{-6}{10}$ = -0.6

The third term of the geometric progression, $a_3$ = 3.6 = $a_2$ × r = -6 × -0.6 = 3.6

The fourth term of the geometric progression, $a_4$ = -2.16 = $a_3$ × r = 3.6 × -0.6 = -2.16

So the above confirms that the series is indeed a geometric progression with first term, $a_{1}$ = 10 and constant ratio, r  = -0.6

Now to find if the series converges or diverges we can take sum of the series from first term to infinity.

We can already infer that the series is convergent since the constant ratio, r  = -0.6 which is less than 1.

So from the formula for the sum to infinity of a geometric progression, S, we get

S = $\frac{a_1}{1 - r} = \frac{10}{1 - (-0.6)} = \frac{10}{1 + 0.6} = \frac{10}{1.6} = 6.25$

So the series is convergent with the sum being 6.25