Question

Determine whether the sequence converges or diverges. If it converges, give the limit.

Answer 1

The sequence diverges because the value of the absolute common ratio r is greater than the 1.

What is convergent of a series?

A series is convergent if the series of its partial sums approaches a limit; that really is, when the values are added one after the other in the order defined by the numbers, the partial sums getting closer and closer to a certain number.

We have series:

9, 27, 81, 243....

The above series is a geometric progression with common ratio r is 3

$\rm r =\dfrac{27}{9}$

r = 3

We know the formula for a geometric sequence:

$\rm S_n = 9(3)^n$

$\rm S_n = 3^{n+2}$

A geometric series converges only if the absolute value of the common ratio:

r < 1  and

It diverges if the ratio ≥ 1

Here the value of r = 3 which is greater than the 1 so the sequence diverges.

Thus, the sequence diverges because the value of the absolute common ratio r is greater than the 1.

Learn more about the convergent of a series here:

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