PLEASE HELP ME!!! Give explanation too! Thank you so much! Match each series to the test that should be used to test for convergence/divergence. While it is possible that each test could apply to more than one series, in this exercise each is only used once. (5 points)
1. the summation from n = 1 to infinity of 2n - 3/n - 1
2. the summation from n = 1 to infinity of (-1)^n n^2/n^3 + 1
3. the summation from n = 1 to infinity of n^2e^-n^3
4. the summation from n = 1 to infinity of 3^n/(n+1)!
a. alternating series test
b. integral test
c. ration test
d. nth term test
To determine which test should be used to test for convergence/divergence of each series, we can analyze the properties of the series and match them to the appropriate test.
1. the summation from n = 1 to infinity of 2n - 3/n - 1:
To analyze this series, we can use the ratio test. The ratio test is used to determine convergence or divergence of a series by taking the limit of the absolute value of the ratio of successive terms. If the limit is less than 1, the series converges; if it is greater than 1, the series diverges; and if it equals 1, the test is inconclusive.
2. the summation from n = 1 to infinity of (-1)^n n^2/n^3 + 1:
For this series, we can apply the integral test. The integral test compares the series to an improper integral. If the integral converges, the series converges; if the integral diverges, the series diverges.
3. the summation from n = 1 to infinity of n^2e^-n^3:
Here, we can use the nth term test. The nth term test states that if the nth term of the series does not approach zero as n approaches infinity, the series diverges. If the nth term approaches zero, then the test is inconclusive, and other tests must be used.
4. the summation from n = 1 to infinity of 3^n/(n+1)!:
To analyze this series, we can use the ratio test. The ratio test is again applicable here, as it helps us determine the convergence or divergence of a series based on the limit of the absolute value of the ratio of successive terms.
Based on the analysis above, we can match each series to the appropriate test as follows:
1. Series 1: Ratio test (c)
2. Series 2: Integral test (b)
3. Series 3: nth term test (d)
4. Series 4: Ratio test (c)
Please note that while these matches are valid for this exercise, it is important to understand that in some cases, multiple tests may be applicable to a given series, and further analysis may be required to confirm the results.