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. (4 points)
1. the summation from n equals 1 to infinity of the quotient of the quantity 2 times n minus 3 and the quantity n minus 1
2. the summation from n equals 1 to infinity of the quotient of n sqaured and the quantity n to the 3rd power plus 1
3. the summation from n equals 1 to infinity of the product of n squared and e raised to negative 1 times n cubed power
4. the summation from n equals 1 to infinity of the quotient of 3 raised to the nth power and n plus 1 factorial
A) alternating series test
B) integral test
C) ratio test
D) nth term test
To match each series to the test that should be used to test for convergence/divergence, we need to understand the different convergence tests available. Here is an explanation of each test:
1. Alternating Series Test:
- Used for alternating series where the terms alternate signs.
- The series is convergent if it satisfies three conditions:
- The terms decrease in absolute value.
- The limit of the terms as n approaches infinity is 0.
- The terms are alternately positive and negative.
2. Integral Test:
- Used when the terms of the series resemble the terms of an improper integral.
- If the integral of the terms converges, the series converges.
- If the integral of the terms diverges, the series diverges.
3. Ratio Test:
- Used when the terms of the series involve exponentials, factorials, or powers of n.
- If the limit of the absolute value of the ratio of consecutive terms is less than 1, the series converges.
- If the limit is greater than 1, the series diverges.
- If the limit is exactly 1, the test is inconclusive.
4. nth Term Test (also called Divergence Test):
- Used as a quick test to check for divergence.
- If the limit of the terms as n approaches infinity is not 0, then the series diverges.
- If the limit is 0 or the test is inconclusive, further tests are needed.
Now, let's match each series to the appropriate test:
1. Series: the summation from n equals 1 to infinity of the quotient of the quantity 2n - 3 and (n - 1)
Test: Ratio Test (C)
2. Series: the summation from n equals 1 to infinity of the quotient of n squared and (n^3 + 1)
Test: Ratio Test (C)
3. Series: the summation from n equals 1 to infinity of the product of n squared and e raised to negative 1 times n cubed
Test: Integral Test (B)
4. Series: the summation from n equals 1 to infinity of the quotient of 3 raised to the nth power and (n + 1) factorial
Test: Ratio Test (C)
So the matching is:
1. Series 1 - Test C
2. Series 2 - Test C
3. Series 3 - Test B
4. Series 4 - Test C
how about you try one now?
These slow-motion homework dumps get old after a while.