State the convergence/divergence for the series the summation from n equals 1 to infinity of the product of negative 1 raised to the nth power and 2 raised to the nth power divided by n factorial . (4 points)
A) Converges absolutely
B) Converges conditionally
C) Diverges
D) Cannot be determined
did you ever figure this out? i'm stuck too
To determine the convergence or divergence of the series, we can use the ratio test.
The ratio test states that for a series ∑(a_n), if the limit of the absolute value of the ratio of consecutive terms is less than 1, then the series converges. If the limit is greater than 1 or does not exist, then the series diverges. If the limit is equal to 1, then the test is inconclusive.
Let's apply the ratio test to the given series:
We have the series: ∑((-1)^n * 2^n / n!)
Let's calculate the ratio of consecutive terms:
(a_(n+1)) / (a_n) = ((-1)^(n+1) * 2^(n+1) / (n+1)!) / ((-1)^n * 2^n / n!)
Simplifying, we can cancel out the common factors:
= ((-1)^(n+1) * 2^(n+1) * n!) / ((n+1)! * (-1)^n * 2^n)
= (-1) * 2 * n! / (n! * (n+1))
= (-2) / (n+1)
As n approaches infinity, the ratio tends to -2.
Since the limit of the ratio is greater than 1, the series diverges.
Therefore, the answer is:
C) Diverges