Determine if series is absolutely convergent, conditionally convergent or divergent.
(-1)^(n-1) *(9^n)/(n^8)
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.
To determine if the series (-1)^(n-1) * (9^n)/(n^8) is absolutely convergent, conditionally convergent, or divergent, we can use the alternating series test and the ratio test.
1. Alternating series test:
Let's first check if the series (-1)^(n-1) * (9^n)/(n^8) satisfies the conditions of the alternating series test:
a) The terms alternate in sign: The series has (-1)^(n-1) as a factor, which alternates between -1 and 1 as n changes.
b) The absolute value of the terms decreases: Let's consider the absolute value of the terms: |(-1)^(n-1) * (9^n)/(n^8)|.
Taking the absolute value eliminates the alternating sign, so we are left with (9^n)/(n^8).
To check if this sequence decreases as n increases, we can evaluate the ratio of consecutive terms:
|[(9^(n+1))/((n+1)^8)] / [(9^n)/(n^8)]| = |(9^(n+1)*n^8) / (9^n*(n+1)^8)|
= |(9n^8) / ((n+1)^8)|
As n increases, the denominator (n+1)^8 increases faster than the numerator 9n^8, which means the sequence decreases in value as n increases.
Since the series satisfies both conditions of the alternating series test, we can conclude that the series is convergent.
2. Ratio test:
Next, let's apply the ratio test to further determine if the series is absolutely convergent or conditionally convergent:
The ratio test states that if the limit of the absolute value of the ratio of consecutive terms is less than 1, the series is absolutely convergent. If it is greater than 1, the series is divergent. If it is exactly 1 or the limit does not exist, the test is inconclusive.
Let's calculate the limit of the ratio of consecutive terms:
lim{n->infinity} |[(-1)^(n-1+1) * (9^(n+1))/((n+1)^8)] / [(-1)^(n-1) * (9^n)/(n^8)]|
= lim{n->infinity} |[(9^(n+1)*n^8) / (9n^8*(n+1)^8)]|
= lim{n->infinity} |(9(n+1)^8) / (9n^8*(n+1)^8)|
= lim{n->infinity} |1 / (n^8)|
As n approaches infinity, the term (1 / (n^8)) approaches 0. Therefore, the limit is less than 1.
Based on the ratio test, the series (-1)^(n-1) * (9^n)/(n^8) is absolutely convergent.
In conclusion, the series (-1)^(n-1) * (9^n)/(n^8) is absolutely convergent.