Consider the following geometric series.

[infinity]∑n=1 (-8) n-19n

a) Find the common ratio.

b) Determine whether the geometric series is convergent or divergent.

c) If it is convergent, find its sum. (If the quantity diverges, answer diverges.)

Question

Mathematics

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6 March, 03:00

Consider the following geometric series.

[infinity]∑n=1 (-8) n-19n

a) Find the common ratio.

b) Determine whether the geometric series is convergent or divergent.

c) If it is convergent, find its sum. (If the quantity diverges, answer diverges.)

+5

Answers (1)

Know the Answer?

Avery Mcpherson · Answer 1

a) - 8/9

b) The series is a convergent series

c) 1/17

Step-by-step explanation:

The series a+ar+ar²+ar³⋯ = ∑ar^ (n-1) is called a geometric series, and r is called the common ratio.

If - 1
a is the first tern of the series.

a) Rewriting the series ∑ (-8) ^ (n-1) / 9^n given in the form ∑ar^ (n-1) we have;

∑ (-8) ^ (n-1) / 9^n

= ∑ (-8) ^ (n-1) / 9• (9) ^n-1

= ∑1/9 • (-8/9) ^ (n-1)

From the series gotten, it can be seen in comparison that a = 1/9 and r = - 8/9

The common ratio r = - 8/9

b) Remember that for the series to be convergent, - 1
c) Since the sun of the series tends to infinity, we will use the formula for finding the sum to infinity of a geometric series.

S∞ = a/1-r

Given a = 1/9 and r = - 8/9

S∞ = (1/9) / 1 - (-8/9)

S∞ = (1/9) / 1+8/9

S∞ = (1/9) / 17/9

S∞ = 1/9*9/17

S∞ = 1/17

The sum of the geometric series is 1/17

Consider the following geometric series.[infinity]∑n=1 (-8) n-19na) Find the common ratio.b) Determine whether the geometric series is convergent or divergent.c) If it is convergent, find its sum. (If the quantity diverges, answer diverges.)

Consider the following geometric series.

[infinity]∑n=1 (-8) n-19n

a) Find the common ratio.

b) Determine whether the geometric series is convergent or divergent.

c) If it is convergent, find its sum. (If the quantity diverges, answer diverges.)