Determine if series is absolutely convergent, conditionally convergent or divergent.

Question

(-1)^(n-1) *(9^n)/(n^8)

Answer 1

To determine if the series is absolutely convergent or divergent, we need to consider the series |(-1)^(n-1) *(9^n)/(n^8)|.

|(-1)^(n-1) *(9^n)/(n^8)| = (9/|n|)^8

Since the series (9/|n|)^8 is a p-series with p = 8, which is greater than 1, it is convergent by the p-series test.

Therefore, the original series (-1)^(n-1) *(9^n)/(n^8) is absolutely convergent.

To determine if the series is conditionally convergent or not, we need to evaluate the limit of the absolute value of the terms of the series.

lim |(-1)^(n-1) *(9^n)/(n^8)| = lim (9/n)^8 = 0

Since the limit of the absolute value of the terms of the series is equal to 0, but the series is not absolutely convergent, it is not conditionally convergent either.

Therefore, the series is absolutely convergent.