Question

Determine if the series {(π) ⁿ/3ⁿ⁺¹}[infinity]ₙ₌₁ is convergent or divergent. If possible, find the sum.

(a) List the first five terms of the sequence.
(b) Determine if the sequence is convergent or divergent. If convergent, find the limit.
(c) List the first five terms of the associated series.
(d) Determine if the series is convergent or divergent. If convergent, find the sum.

Answer 1

The sequence converges to 0 and the series is convergent.

Step-by-step explanation:

To determine if the series {(π) ⁿ/3ⁿ⁺¹}[infinity]ₙ₌₁ is convergent or divergent, we can first list the first five terms of the sequence:

1. (π/3)(π/9)
2. (π/9)(π/9)
3. (π/9)(π/9)(π/9)
4. (π/9)(π/9)(π/9)(π/9)
5. (π/9)(π/9)(π/9)(π/9)(π/9)

Next, we can determine if the sequence converges or diverges. In this case, as n approaches infinity, the terms of the sequence get smaller and smaller. Therefore, the sequence converges to 0.

Now, let's list the first five terms of the associated series:

1. (π/3)(π/9)
2. (π/9)(π/9)(π/9)
3. (π/9)(π/9)(π/9)(π/9)(π/9)
4. (π/9)(π/9)(π/9)(π/9)(π/9)(π/9)(π/9)
5. (π/9)(π/9)(π/9)(π/9)(π/9)(π/9)(π/9)(π/9)(π/9)

Finally, we can determine if the series is convergent or divergent. Since the terms of the series approach 0 as n approaches infinity, the series is convergent. However, finding the exact sum of the series requires further calculations.