Question

Test the series for convergence or divergence using the Alternating Series Test.

−2 8 + 4 9 −6 10 + 8 11 −10 12 +
Identify
bn.
(Assume the series starts at n = 1.)
Evaluate the following limit.
lim n→[infinity] bn
Since
lim n→[infinity] bn
(select) = or ≠ 0 and
bn + 1 select (≤ or ≥) bn
for all n, (-Select)the series converges OR the series diverges OR the test is inconclusive .

Answer 1

To test the convergence or divergence of the given series using the Alternating Series Test, identify bn as
`-1^((n+1)/n)` and evaluate the limit as n approaches infinity. The series converges.

Step-by-step explanation:

To test the convergence or divergence of the given series, we can apply the Alternating Series Test. The series can be written as:

−2/8 + 4/9 − 6/10 + 8/11 − 10/12 + ...

To identify bn, we can observe that the even terms (starting with the first term) have a positive sign and the odd terms have a negative sign. So, bn =
`-1^((n+1)/n)`

Next, we need to evaluate the limit as n approaches infinity. lim n→∞ bn = lim n→∞
`-1^((n+1)/n)` = 0.

Since lim n→∞ bn = 0 and bn+1 ≤ bn for all n, the Alternating Series Test tells us that the series converges.

Answer 2

To test the series for convergence or divergence using the Alternating Series Test, we need to identify the sequence of terms and evaluate the limit of the sequence.

Step-by-step explanation:

To test the series for convergence or divergence using the Alternating Series Test, we need to identify the sequence of terms and evaluate the limit of the sequence. The given series is:

-2/8 + 4/9 - 6/10 + 8/11 - 10/12 + ...

From the pattern, we can see that bn is the sequence of numerators: 2, 4, 6, 8, 10, ...

To evaluate the limit of the sequence, we take the limit as n approaches infinity:

lim (n->∞) bn = 2

Since the limit (2) is not equal to 0, and the terms of the series alternate in sign but do not eventually approach 0, the test is inconclusive. Therefore, we cannot determine whether the series converges or diverges using the Alternating Series Test.