Determine the convergence or divergence of the series. Indicate the test that was used and justify your answer.
Sigma (lower index n = 1; upper index infinity) [(-1)^n*3^n]/(n*2^n)
To determine the convergence or divergence of the given series, we can use the ratio test. The ratio test can be applied to a series of the form ∑(a_n), where a_n is a sequence of non-zero numbers.
The ratio test states that if the limit of the absolute value of the ratio of consecutive terms, lim as n approaches infinity of |(a_n+1)/a_n|, is less than 1, then the series converges absolutely. If the limit is greater than 1 or does not exist, then the series diverges. If the limit is equal to 1, the test is inconclusive.
Now, let's apply the ratio test to the given series:
a_n = (-1)^n * 3^n / (n * 2^n)
We need to find the limit of |(a_n+1)/a_n| as n approaches infinity:
|(a_n+1)/a_n| = |((-1)^(n+1) * 3^(n+1) / ((n+1) * 2^(n+1))) / ((-1)^n * 3^n / (n * 2^n))|
Simplifying the expression, we get:
|(a_n+1)/a_n| = [(3 * 2^n) / (2 * n+1)]
Now, let's take the limit of this expression as n approaches infinity:
lim as n approaches infinity of [(3 * 2^n) / (2 * n+1)] = ∞
Since the limit is greater than 1, the ratio test tells us that the series diverges.
Therefore, the series ∑ [(-1)^n * 3^n] / (n * 2^n) is divergent, as determined by the ratio test.